nonreversible homoclinic snaking scenarios€¦ · solutions that exist in a neighbourhood of what...

Technische Universitat Ilmenau

Fakultat fur Mathematik

und Naturwissenschaften

Fachgebiet Analysis und Systemtheorie

Nonreversible Homoclinic Snaking Scenarios

Dissertation zum Erlangen des akadem. Grades Dr. rer. nat.

Martin Vielitz

Betreuer:PD Dr. Jurgen Knobloch

Gutachter:PD Dr. Jurgen KnoblochProf. Dr. Bernold FiedlerProf. Dr. Bjorn Sandstede

Eingereicht: Ilmenau, den 11.07.2014

Wissenschaftliche Aussprache: Ilmenau, den 21.11.2014

urn:nbn:de:gbv:ilm1-2014000391

i

Gegen Zielsetzungen ist nichts einzuwenden,

sofern man sich dadurch nicht von interessanten Umwegen abhalten laßt.

Mark Twain

CONTENTS iii

Contents

1 Introduction 1

2 Preliminaries 13

2.1 Unperturbed snaking – snakes and ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Fenichel coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 The solution of the Shilnikov problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Nonreversible perturbations 23

3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Continuation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 The determination equation for 1-homoclinic orbits . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.2 Continuation in the perturbed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.1 Connection with Poincare maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.2 Transition from isolas to criss-cross snaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 General perturbations 37

4.1 Setup and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Continuation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 The determination equation for 1-homoclinic orbits . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.2 Continuation near curve segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.3 The bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Nonreversible snaking 55

5.1 Setup and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Snaking analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3 Negative Floquet multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.4 Isolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 The construction of Fenichel coordinates 75

6.1 A foliation of W e,uuloc (P ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.1.1 Preliminaries and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.1.2 A moving coordinate system near P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.1.3 Foliations of W e,uuloc (P ) and W ss,e

loc (P ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.1.4 Reversibility and dependency on parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2 The vector field in the Fenichel coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7 The Shilnikov problem 99

7.1 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.2 The reference solutions q± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3 The linearised equation along q± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4 The solutions of the inhomogeneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.5 The fixed point equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.6 The estimates of the solutions X± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8 Discussion and conclusions 131

Bibliography 138

1

Chapter 1

Introduction∗

The points of the leopard, the shape of the snowflakes, the structure of the honeycombs– all these are examples for

complex patterns that appear everywhere in nature. But also many physical [5], biological [45], optical [1] or chemical

processes [53] reveal spatially localised patterns. It is no wonder that the study of localised patterns has attracted

much attention across several disciplines, not only to understand the complex behaviour of the modelling system, but

also as an object in its own right.

In recent years numerous experiments have exposed the existence of localised patterns in several partial differential

equations (PDEs) modelling diverse processes. These localised patterns are special states that are ”embedded“ in a

”background state“. If the spatial state of the considered PDE is 1-dimensional, then localised patterns of the PDE

may be described by solutions of an ordinary differential equation (ODE). More precisely, if the localised patterns are

embedded in the background of the steady state and if the patterns are time-independent or are travelling waves, then

these patterns correspond to homoclinic solutions (to an equilibrium) of the associated ODE. Hence the study of the

localised patterns of the PDE is closely related to the study of homoclinic orbits in the ODE. Most often the related

ODE is reversible and Hamiltonian. If the PDE depends additionally on parameters, it arises the question at which

parameter values localised patterns do exist. In terms of the ODE this is a continuation problem of homoclinic orbits.

Several works have investigated numerically continuation problems of homoclinic orbits that are derived from pattern

forming PDEs, see for example the references in [3]. In some of those works a very special continuation scenario,

so-called Homoclinic Snaking, was observed.

As a specific example we consider the Swift-Hohenberg equation [69], which became a sort of standard model PDE

for the phenomenon of Homoclinic Snaking. More precisely we consider the 1-dimensional quadratic-cubic Swift-

Hohenberg equation

ut = µu− (1 + ∂2x)2u+ 2u2 − u3, (1.1)

with µ as continuation parameter or snaking parameter, respectively. The steady states of that PDE satisfy the 4th

order ODE

uxxxx = (µ− 1)u− 2uxx + 2u2 − u3. (1.2)

The Swift-Hohenberg equation is reversible, i.e. it is invariant under the symmetry x→ −x. This structure is inherited

by the ODE (1.2). Additionally, Equation (1.2) conserves the first integral

F (u) = −1

2(µ− 1)u2 + u2x −

1

2u2xx + uxuxxx −

2

3u3 +

1

4u4.

In fact, (1.2) can even be written as a Hamiltonian system, see [72]. Equation (1.2) has a multitude of homoclinic

∗ The introduction contains parts of the introduction in [37] and [38].

2 1 Introduction

solutions that exist in a neighbourhood of what we call an EtoP cycle. This is a heteroclinic cycle connecting an

equilibrium E and a periodic orbit P . Here we are interested in so-called 1-homoclinic orbits, that are orbits, which

follow the EtoP cycle just once. (In contrast to N-homoclinic orbits that follow the EtoP cycle N times.) In what

follows we address 1-homoclinic orbits just as homoclinic orbits.

According to the first integral, the homoclinic orbits to E are robust, meaning that they will not be destroyed

under (small) perturbations. Consequently, such orbits can be continued within a 1-parameter family of differential

equations. For Equation (1.2) this parameter is µ and the corresponding bifurcation diagram is displayed in Figure 1.1.

As remarked, Equation (1.2) exhibits the effect of Homoclinic Snaking [9, 72], which we explain now in more detail.

In the bifurcation diagram (Figure 1.1) there are two blue intertwining curves. These curves correspond to symmetric

homoclinic orbits of (1.2), that are orbits, which are invariant under the reversible symmetry. These curves are called

snaking curves. We observe that along the two snaking curves the L2 norm of the related homoclinic orbits increases.

This is due to the process that by following the snaking curves upwards, the corresponding homoclinic orbits perform

more and more windings about the periodic orbit P , which results in a growing orbit length. The turning points of

the blue snaking curves indicate saddle-node bifurcations of the symmetric homoclinic orbits. The snaking curves are

connected by red lines, which we call rungs (of the ladder). The rungs are related to asymmetric homoclinic orbits, see

panels (c), (d). The asymmetric homoclinic orbits bifurcate from the symmetric ones via pitchfork bifurcation. These

bifurcation points are close to the saddle-nodes of the snaking curves of symmetric homoclinic orbits [3]. Additionally,

but not displayed in Figure 1.1, one can consider the bifurcation diagram of the heteroclinic EtoP cycle. Roughly

speaking, the µ-range of the snaking curves is the µ-range for which the EtoP cycle does exist. Close to the endpoints

of the µ-interval the involved EtoP and PtoE connections (note that they are images of each other by the reversing

symmetry) simultaneously solve in the course of a saddle-node bifurcation. The overall structure in this bifurcation

diagram is often called snakes-and-ladders structure.

0.50 0.45 0.40 0.35 0.30

0.10

0.15

0.20

0.25

0.30

40 0 40

0.5

0.5

1.5

40 0 40

0.5

0.5

1.5

40 0 40

0.5

0.5

1.5

40 0 40

0.5

0.5

1.5

2L

(a)(c,d)

(b)

(a) (b)

(c) (d)

µ

u(x)

Figure 1.1: Homoclinic Snaking in the Swift-Hohenberg equation (1.2). [38, Figure 1]

The computational results in Figure 1.1 originate from [37] and have been obtained using AUTO [17]. Bifurcation

diagrams as displayed in Figure 1.1 have been discussed for instance in [3, 8, 14, 72].

3

Homoclinic Snaking has been investigated analytically in all details the first time by Beck et al. in [3]. Starting

straightaway in the context of ODEs, the authors consider a smooth family of differential equations

x = f(x, µ), x ∈ R4, µ ∈ R, (1.3)

which is reversible and has a first integral for all values of the snaking parameter µ. It is assumed that there is an

EtoP cycle connecting an equilibrium E and a (restricted to a level set hyperbolic) periodic orbit P . The authors

formulate a set of hypotheses about this EtoP cycle that guarantees a snakes-and-ladders structure as in Figure 1.1.

In the course of their investigations the authors show that this snakes-and-ladders structure is related to the behaviour

of the intersection of the unstable manifold of E and the stable manifold of P . The intersection of these manifolds is

described by a certain function Z0 and it turns out that homoclinic orbits near the EtoP cycle are related to the zero

level set of Z0. The relation of the function Z0 and the bifurcation diagram is illustrated in Figure 1.2. Furthermore

the analysis in [3] relies on the existence of Fenichel coordinates near a hyperbolic periodic orbit, as well as, a solution

of a Shilnikov problem near a periodic orbit. A more detailed overview of the results of [3] is given in Section 2.1.

ϕ

π 2π0

L L

µ

L0M (n+ 1)

LπM (n)

L0m(n)

Lπm(n)

L0M (n)

Qπ−(n+ 1)Q0+(n+ 1)

Qπ+(n)Q0−(n)

Qπ−(n)Q0+(n)

Figure 1.2: The left panel shows the zero level set of Z0, and the right panel the snakes-and-ladders structure. Verticallines on the left correspond to the snakes, and the horizontal curves correspond to the rungs in Figure 1.1. The levelset is reflection-symmetric with respect to the line ϕ = π and moreover 2π-periodic in ϕ, and so the outer vertical

lines represent the same snaking curve. The lower index +/− indicates that within the corresponding region Q0/π

+/−

the function Z0 is positive/negative. [38, Figure 3]

The aim of this thesis is twofold. On the one hand we consider perturbations of the snaking scenarios in [3]. In the

course of this we distinguish perturbations that destroy only the reversing symmetry, Chapter 3, and perturbations

that destroy both, the reversing symmetry and the first integral, Chapter 4. Further we consider Homoclinic Snaking

in ODEs that have no particular structure from the beginning, Chapter 5. On the other hand we adapt the analysis of

the above mentioned Fenichel coordinates, Chapter 6, and a Shilnikov problem to our particular situation, Chapter 7.

4 1 Introduction

(b)(a)

Figure 1.3: A pitchfork bifurcation (a) and its unfolding (b).

ϕ

L(a)

ϕ

L(b)

Figure 1.4: The left part in each panel shows the zero level set of Z as a perturbation of the zero level set of Z0. Theright part in each panel shows the corresponding continuation curves for nearby homoclinic orbits as perturbationsof the snaking scenario in Figure 1.1. [38, Figure 4]

In what follows we describe the single problems which we consider and our main results in more detail.

Regarding the addressed perturbation problems, we consider a smooth family of differential equations,

x = f(x, µ, λ), x ∈ R4, (µ, λ) ∈ R2,

where µ plays again the role of the snaking parameter and λ acts as a perturbation parameter. We claim that for

λ = 0 the hypotheses of [3] are valid. In particular, when λ = 0, then the periodic orbit is hyperbolic within the

corresponding level set of the first integral. Furthermore, E and P are symmetric and therefore the EtoP cycle is

symmetric. For λ 6= 0 the structural assumptions of Equation (1.3) are perturbed. We formulate this perturbation in

the terms of the intersection of the manifolds of E and the manifolds of P . We show that similarly to the considerations

in [3] the continuation curves are closely related to the zero level set of a certain function Z = Z(λ), where Z(0) = Z0.

We consider the following two perturbation scenarios:

In Chapter 3 we investigate the effects when λ causes a breaking of the reversible symmetry, while the conservative

character of the ODE remains. Homoclinic solutions will persist under such perturbations, since restricted to the

3-dimensional level set the EtoP cycle is robust. But because the reversible symmetry is broken the pitchfork bifurca-

tions in the bifurcation diagram will unfold, as displayed in Figure 1.3. That means the two snaking curves (belonging

to the symmetric homoclinic orbits) separate at the pitchfork bifurcations, which alters the overall bifurcation diagram.

5

Now the question is how the single branches connect globally. It turns out that two different scenarios are possible:

In the first scenario the zero level set of the function Z consists of closed curves as it is depicted in Figure 1.4, panel

(a). In the bifurcation diagram two pieces of one (now separated) snaking curve connect with two asymmetric branches

(the primary rungs of the ladder) and form a figure-eight shaped, closed curve, called isola. In this case the bifurcation

diagram consists of stacked isolas.

In the second scenario the zero level set of Z has a form as depicted in Figure 1.4, panel (b). There the corresponding

continuation curves follow alternately the original (thin blue) continuation curves of symmetric orbits and in the

transition between these curves they follow the primary (thin red) rungs. We call this scenario criss-cross snaking.

The results of Chapter 3 are summarised in Theorem 3.1.

The described scenarios can also be ”validated“ numerically. As an example for the first scenario let us consider the

following ODE (which is also derived from the Swift-Hohenberg equation):

uxxxx = (µ− 1)u− 2uxx + 2u2 − u3 + λ(

3uxu2xx + u2xuxxx

)

. (1.4)

This ODE conserves the function

Fλ(u) = F (u) + λu3xuxx,

but it is not reversible for λ 6= 0. Panel (a) in Figure 1.5 shows the bifurcation diagram of (1.4) for λ = 0.3. We see

that the snakes-and-ladders structure breaks up into a family of stacked isolas.

0.50 0.45 0.40 0.35

0.10

0.15

0.20

0.25

0.30

0.50 0.45 0.40 0.35

0.10

0.15

0.20

0.25

0.30

(a) (b)

L2

L2

µ µ

Figure 1.5: Isolas of homoclinic solutions of Equation (1.4) with λ = 0.3 in panel (a) and criss-cross snaking ofhomoclinic solutions of Equation (1.5) with λ = 0.3 in panel (b). The middle part contains magnifications of partsof the diagrams as indicated. [38, Figure 2]

6 1 Introduction

An example for criss-cross snaking gives the following equation

uxxxx = (µ− 1)u− 2uxx + 2u2 − u3 + 3λuxuxx, (1.5)

which possesses the first integral

Fλ(u) = F (u) + λu3x.

Again, the perturbation breaks the reversibility, if λ 6= 0. As described above the homoclinic solutions still lie on

snakes in the bifurcation diagram. In contrast to the situation for Equation (1.4), the perturbed snaking curves follow

the complete snakes-and-ladders structure including the rungs.

In Chapter 4 we consider perturbations that destroy both the reversible structure and the first integral. Motivation

for these investigations are the numerical considerations in [7]. Again the setup in [3] yields the basis of our analysis.

This means that (as in Chapter 3) we assume in the unperturbed case the presence of an EtoP cycle and describe

the character of the perturbation in terms of the involved manifolds of E and P . Though the approach in Chapter 4

is similar to the one in Chapter 3, there are several differences that distinguish both situations. First of all the loss

of the reversible structure and the first integral effects that the intersection of the manifolds of E and P takes place

in a further, additional dimension. Thus these manifolds do no longer intersect transversally and the corresponding

heteroclinic orbits of the EtoP cycle become codimension-1 objects. Consequently they are not robust and the nearby

homoclinic orbits cannot (generically) be continued within a 1-parameter family.

Furthermore, the dynamic near the periodic orbit differs qualitatively in the perturbed and unperturbed case. As

long as no perturbation is present, the periodic orbit possesses a 1-dimensional (strongly) stable and a 1-dimensional

(strongly) unstable manifold. By perturbing the reversible symmetry and the first integral, the dimension of one of

these manifolds (in our setup the stable one) changes.

Our main result regarding this class of perturbations is summarised in Theorem 4.1. We show that the continuation

takes place in two parameters, the snaking parameter µ and the perturbation parameter λ. We find that homoclinic

orbits exist along a curve (µ(L), λ(L)) in parameter space. This is in contrast to the purely nonreversible perturbations

in Chapter 3, where λ remains constant during the continuation. Recall that the parameter L corresponds to the time

the homoclinic orbit spends near P and that this quantity is strongly related to the L2 norm of the homoclinic orbit.

It turns out that λ(L)→ 0 as L→∞, whereas the diagram L versus µ(L) looks like the drawings in Figure 1.4 panel

(a) or panel (b), respectively. However, note that the L versus µ(L) plot is merely a projection of the full bifurcation

diagram, which comprises also the parameter λ. We refer to Section 4.3 for a discussion of our analytical results and

numerical observations in the motivating example [7].

In Chapter 5 we consider Homoclinic Snaking in 3-dimensional ODE systems without particular structure. Motivation

for these considerations is an example by Krauskopf, Oldeman and Rieß [42, 43]. The authors have considered a

family of vector fields in R3 without particular structure such as reversible symmetries or a Hamiltonian structure.

Interestingly also in this ODE the authors have numerically discovered an effect that is very similar to the Homoclinic

Snaking scenarios in the Swift-Hohenberg equation that we have discussed so far. The investigated family of vector

7

fields with family parameter (µ1, µ2) reads

x = µ1x− y + x sinϕ− (x2 + y2)x+ 0.01(2 cosϕ+ µ2)2

y = µ1y + x+ y sinϕ− (x2 + y2)y + 0.01π(2 cosϕ+ µ2)2

ϕ= µ2 − (x2 + y2) + 2 cosϕ

=: F (x, y, ϕ, µ1, µ2). (1.6)

Equation (1.6) has a hyperbolic equilibrium E with a 1-dimensional stable manifold W s(E). Further, (1.6) possesses a

saddle periodic orbit P , meaning that both, the stable and the unstable manifold of P are 2-dimensional. Again there

exists a cycle between E and P . According to the dimensions of the involved manifolds, the heteroclinic connection

between E and P will generically be robust, while a connection between P and E will be of codimension-1. We refer

to Figure 1.6 for a sketch of the EtoP cycle. In contrast to the examples (1.2) and (1.4), the existence of homoclinic

P

E

Figure 1.6: Sketch of the EtoP cycle of (1.6) in a suitable coordinate system. [37, Figure 2].

orbits is not generic, since the stable and unstable manifolds of E will typically not intersect by the Kupka-Smale

theorem [59]. Transversality arguments show that in 1-parameter families of differential equations one can expect

an intersection of these manifolds, and hence the occurrence of a homoclinic orbit at an isolated parameter value.

Therefore a continuation of homoclinic orbits can be carried out in the two parameters (µ1, µ2). Figure 1.7 displays a

continuation curve hb1 for a homoclinic orbit that has been detected numerically in [42, 43]. Again, and not only due

to its shape, we address this curve as snaking curve.

As in the reversible case, the homoclinic orbit under consideration is 1-homoclinic w.r.t. the EtoP cycle and along

the continuation curve the homoclinic orbits perform more and more windings about the periodic orbit – see panels

(b)-(d) in Figure 1.7, which shows plots of one state variable corresponding to points indicated in panel (a). The plot

of the L2-norm of the (x, y)-part of the solution versus the parameter µ1 or µ2, respectively, behaves as in the original

Homoclinic Snaking scenario, see Figure 1.8. The snaking behaviour w.r.t. both parameters is due to the declination

of the curve cb in Figure 1.7 (a). This defines intervals within which the parameters move while hb1 approaches cb. We

refer to [42, 43] and [37,58] for more numerical results regarding this system.

Although system (1.6) has the same dimension as the restriction of the above original Homoclinic Snaking system to

a level set, the geometry is quite different.

Here one of the heteroclinic connections constituting the EtoP cycle does not lie in a transversal intersection of the

corresponding stable and unstable manifolds. Hence, by the same arguments as given above for homoclinic orbits,

one can expect to find this heteroclinic connection on a continuation curve in the (µ1, µ2)-space. This is the curve

cb in Figure 1.7 (a). The other connecting orbit is robust – as the ones in the Hamiltonian case. This connecting

orbit exists within the stripe delimited by the curves tb, see again Figure 1.7 (a). In other words, the region where the

snaking curve is located is related to the existence of the robust heteroclinic connection between the equilibrium and

8 1 Introduction

Figure 1.7: Snaking diagram of a 3-dimensional laser model (1.6). Panel (a) shows the snaking curve hb1 together

with the continuation curve cb of the PtoE connecting orbits and the locus tb of saddle-nodes of the EtoP connectingorbits. Panels (b) - (d) show y vs. time plots of the y component at the corresponding points (b) - (d) in panel (a).Here ν1 and ν2 correspond to µ1 and µ2 in our notation. [37, Figure 3]

the periodic orbit. Remarkably, the snaking curve accumulates at the curve segment defined by the intersection of the

curve cb with the stripe delimited by the curves tb. In other words, the snaking curve accumulates at the line segment

for which the EtoP cycle does exist.

The purpose of Chapter 5 is to verify the numerical results of [42, 43] analytically. In the spirit of the setup in [3]

and Chapters 3 and 4 of this thesis we give a series of hypotheses that are sufficient for Homoclinic Snaking in ODEs

without specific structure. Again we assume the presence of a heteroclinic EtoP cycle. As explained above this cycle

is generically not robust. More precisely it is a codimension-1 object. But we assume that there is an interval I2 such

that for all family-parameter (µ1, µ2) ∈ 0 × I2 the EtoP cycle does exist.

In the course of our analysis we distinguish 2 cases, depending on the sign(s) of the Floquet multipliers of the periodic

orbit. For positive Floquet multipliers our main snaking results, see Theorem 5.1, says that all homoclinic orbits near

the primary EtoP cycles lie on one continuation curve. Roughly speaking this curve can be (again) parametrised by

Figure 1.8: Snaking diagrams of a 3-dimensional model (1.6). Shown are plots of the L2-norm of (x, y) vs. ν1 and ν2(µ1 and µ2 in our notation), respectively, along the snaking curve hb

1. [37, Figure 4]

9

the time the homoclinic orbits spends within a vicinity of P . We show that with increasing time this curve accumulates

at 0 × I2. Observe that this is in keeping with the original Homoclinic Snaking setting, where by going upwards

the snaking curves in the bifurcation diagram the time that the homoclinic solutions spend near the periodic orbit

increases, as well.

For negative Floquet multipliers Theorem 5.2 contains the snaking result. Indeed, the snaking behaviour depends on

the sign of these multipliers. If they are positive, the local (un)stable manifold of P is topologically a cylinder, while

for negative multipliers these local manifolds are topologically a Mobius strip. Negative Floquet multipliers result in

the existence of two snaking curves approaching 0 × I2 from different sides.

Finally we consider a scenario, where the homoclinic orbits are not all located on one continuation curve, but there

exists a sequence of closed homoclinic continuation curves (isolas) in the µ-plane accumulating at 0 × I2, see Theo-

rem 5.3.

As in the context of [3] our analysis relies on the existence of a certain coordinate system near P , so-called Fenichel

coordinates, which we use to formulate our setup. One characteristic of such coordinates is that they reflect the struc-

ture of the (centre) stable and (strong) unstable manifolds near P . Similar coordinates has been used in [3]. There the

presence of the first integral is exploited to obtain the existence of the Fenichel coordinates. Recall that in this thesis

we consider also perturbations of the scenario in [3] that destroy the first integral, cf. Chapter 4. Therefore we have

to establish the Fenichel coordinates in a more general context. More precisely, we have to cope with two additional

difficulties namely: When losing the first integral, we have to take one more space dimension into consideration. And

according to the characteristic of our considered perturbations the weakly stable Floquet exponent tends to zero as

the perturbation tends to zero. Therefore the dynamics near P are qualitatively distinguished in the perturbed and

unperturbed case. Floquet exponents taking these particular conditions into account are established in Chapter 6 and

the results are summarised in Theorem 2.3.

A further main building block for homoclinic orbits near the EtoP cycle are the solutions of a corresponding Shilnikov

problem near P . Also here we adapt the analysis of [3] to our context, taking the above characteristic into account.

The main result in this respect is given in Theorem 2.4 and the corresponding analysis is done in Chapter 7.

Related work:

Homoclinic Snaking especially in the Swift-Hohenberg equation has been analysed in several works, see e.g. [6,9,11,15,

29,49,64], reaching from Homoclinic Snaking on finite or periodic domains in the Swift-Hohenberg equation [4] to the

comparison of Homoclinic Snaking in model equations and physical experiments [23]. In the framework of this thesis

it is not possible to present all the different branches of this field. Here we want to discuss only a selection of those

results that are directly related to our considerations. These are contributions regarding the analysis of Homoclinic

Snaking scenarios, including perturbations of the classical Snaking as depicted in Figure 1.1, as well as, papers regard-

ing numerical studies of perturbations of Homoclinic Snaking scenarios, which mainly motivated our considerations.

The basis for our work is provided in [3], where an analysis of the snaking behaviour in the unperturbed system has

been discussed in detail. A crucial assumption in that respect is the existence of a primary EtoP cycle, which serves

as an organising structure. For the Swift-Hohenberg equation the existence of heteroclinic connections between an

10 1 Introduction

equilibrium and a periodic orbit has been investigated analytically in [11, 41]. In [39, 40] Knobloch and Wagenknecht

have studied Homoclinic Snaking (also called collapsed snaking) near cycles between two equilibria (EtoE cycles)

in 1-parameter families of reversible systems. A geometric explanation of Homoclinic Snaking that considers the

involved manifolds in an appropriated Poincare section has been given in [72, Section 5]. In Section 3.3.1 we briefly

discuss our results using this method. Regarding Chapter 3 we refer to the work by Sandstede and Xu [63], who

have considered snaking scenarios in conservative, nonreversible systems. Similar to our results Sandstede and Xu

have found isolas and snaking continuation curves of homoclinic orbits in those systems. Our emphasis however, lies

on the consideration of nonreversible perturbations, which allows us to describe the corresponding isolas and snaking

continuation curves against the background of the original snakes-and-ladders structure. Further we remark that

in [3, Section 6.4] Homoclinic Snaking in reversible systems without a first integral have been discussed very briefly. It

turns out that due to the reversible structure the two snaking curves are still present in the bifurcation diagram. But

the asymmetric homoclinic orbits cannot be continued in one parameter. So the rungs corresponding to the asymmetric

solutions do no longer appear. However, in [3] it is pointed out that by using a wave approach the asymmetric rungs

can be continued in two parameters, the snaking parameter and the drift speed, cf. also [7].

Many physical systems that exhibit snaking behaviour possess beside a reversing symmetry an additional Z2-symmetry.

Again a prominent example is the Swift-Hohenberg equation. Recently Makrides and Sandstede [48] have studied the

effects of perturbations of homoclinic snaking scenarios that destroy this Z2-symmetry. As in this thesis they have used

the setup of [3] as starting point for their investigations. Applying a ”formal gluing approach of fronts and backs“ the

authors are able to predict the shape of the perturbed bifurcation diagrams. Their results coincide with the numerical

computations in [29] (see below), which shows the universal applicability of the approach in [3].

The behaviour of symmetric 2-homoclinic orbits in the unperturbed system has been analysed in [35]. It turns out

that the corresponding determination equation for these orbits can also be seen as a perturbation of the zero level set

of the function Z0.

In [10], amongst others, Homoclinic Snaking caused by an EtoP cycle in systems in R3 has been considered, by using

a combination of analytical and geometrical arguments. This has been done for the cases where P has positive or

negative Floquet multipliers. These considerations are related to our investigations in Chapter 5, where we give a

rigorous analytical study of those scenarios.

In [36] a Lin’s method approach has been extended to treat heteroclinic chains involving period orbits. These results

are applied to EtoP cycles, in particular to detect nearby 1-homoclinic orbits. However, the results are more local in

nature. In [54,55] Rademacher has applied a somewhat different approach (in handling the flow near P ) of Lin’s method

to study EtoP cycles of codimension-1 and codimension-2. His results about homoclinic orbits near codimension-1

EtoP cycles are of the same nature as the ones in [36].

In [7] a perturbation of the quadratic-cubic Swift-Hohenberg equation that destroys both, the reversibility and the

first integral, has been studied numerically, see also Section 4.3. The authors have shown that under this perturbation

the snakes-and-ladders structure breaks into isolas. This work is one motivation for the studies in Chapter 4 and its

numerical results coincide in many points with the effects of our analytical investigations.

A perturbation of the cubic-quintic Swift-Hohenberg equation has been studied in [29]. The perturbation terms destroy

the Z2 symmetry, but keep the reversing symmetry. In the perturbed bifurcation diagram the primary rungs transform

into S- and Z-shaped curves, whereas the snaking curves are still present.

Finally we remark that our analysis relies on two concepts: Firstly we use Fenichel coordinates near the periodic orbit

(cf. Theorem 2.3 and Chapter 6). These coordinates are based on work by Fenichel [18–21], who has established several

11

Theorems about invariant manifolds. More precisely, we exploit a foliation of a certain extended stable manifold of

P to construct these coordinates. In [28] such foliations have been derived for manifolds corresponding to homoclinic

orbits. In [30–32] Fenichel coordinates has been constructed in the context of slow/fast systems. Though in another

context, the constructions there proceed to a large extent similarly to our considerations. Secondly we exploit a

solution of a Shilnikov problem near P . Such solutions have been derived for example in [32,44,67,68]. Here we adapt

the prove in [44], where a Shilnikov problem has been solved in the context of slow/fast systems. The advantage of

this approach is that not only an estimate of the solutions is derived, but also the leading order terms.

12 1 Introduction

13

Chapter 2

Preliminaries

2.1 Unperturbed snaking – snakes and ladders

In this section we present some of the results of the article “snakes and ladders” [3], where unperturbed Homoclinic

Snaking has been investigated analytically. Here we give an overview of the main results of that work and we presented

only those that are directly related to the considerations in the following chapters. In [3] the authors consider equation

x = f(x, µ), x ∈ R4, µ ∈ Jµ ⊂ R, (2.1)

where Jµ is some interval. They formulate a series of hypotheses under which Equation (2.1) exhibits Homoclinic

Snaking. The hypotheses of [3] are the basis for the setup in Chapter 3 and Chapter 4. In what follows we repeat the

hypotheses of [3] that affect the considerations of this work. At some points we adjust the formulation of the assertion

so that they fit into they setup of Chapter 3 and Chapter 4. At other points we drop some of the assumptions, when

they correspond to investigations that are not related to our problems.

It is assumed that f(x, µ) is a Ck-smooth family of vector fields, where k ∈ N is sufficiently large. The vector field f

is reversible:

Hypothesis (H2.1). There exists a linear involution R : R4 → R4, i.e. R2 = id, such that

f(Rx, µ) = −Rf(x, µ), ∀x ∈ R, µ ∈ Jµ.

The involution R is sometimes called reverser. A solution x(t) of a reversible differential equation is called symmetric,

if the corresponding orbit is invariant under R, that is R(x(t)) = x(t). A solution x(t) is symmetric if, and only if,

there is a τ such that x(τ) ∈ Fix(R). Hence R(x(τ + t)) = x(τ − t). For a more detailed discussion of the properties of

reversible vector fields see e.g. [22,46]. Next the existence of a first integral, a hyperbolic equilibrium and a hyperbolic,

periodic orbit is assumed.

Hypothesis (H2.2). Equation (2.1) has a smooth first integral H : R4× Jµ → R with H = H(x, µ) and H(Rx, µ) =

H(x, µ).

Hypothesis (H2.3). The origin E := x = 0 is a hyperbolic equilibrium of (2.1). More precisely, f(0, µ) = 0 for all

µ and Dxf(0, µ) has two eigenvalues with strictly negative real part and another two eigenvalues with strictly positive

real part. Furthermore E belongs to the zero level set of H, H(E, µ) = 0.

14 2 Preliminaries

Hypothesis (H2.4). For all µ ∈ Jµ Equation (2.1) possesses a periodic orbit P = γt := γ(t, µ) with minimal

period 2π, that satisfies the following:

(i) The family γt depends smoothly on µ ∈ Jµ.

(ii) γ(0, µ) ∈ Fix(R) for all µ ∈ Jµ.

(iii) H(γ(t, µ), µ) = 0 and DxH(γ(t, µ), µ) 6= 0 for all t ∈ R, µ ∈ Jµ.

(iv) Beside the Floquet multiplier 1, the periodic orbit γt possesses the three Floquet multipliers

e−2πα < 1 < e2πα ∀µ ∈ Jµ.

Hypothesis (H2.4) (ii) implies that the periodic orbit P is symmetric. Since the period of P is 2π, it follows that

γ(π, µ) ∈ Fix(R). Further the fixed space Fix(R) = x ∈ R4 |R(x) = x is 2-dimensional, dimFix(R) = 2. The first

assumption of Hypothesis (H2.4) (iii) defines the zero level set of H. The assumption about the derivatives is a regu-

larity condition that implies that the zero level set of H is locally around the periodic orbit a (3-dimensional) manifold.

Further we remark that generically the periodic orbit γ(t, µ) is embedded in a 1-parameter family of periodic orbits.

In fact the existence of the first integral implies that P possesses two Floquet multipliers 1. The first trivial multiplier

corresponds to the vector field direction. The second is related to the first integral. Hence the Implicit Function The-

orem typically guarantees that periodic orbits appear in a 1-parameter family (compare Hypothesis (H2.4) (iv)). A

thorough discussion of this (in the more general framework of vector fields on manifolds) can be found in [66, Theorem

4]. Moreover this family forms locally a smooth, 2-dimensional (centre) manifold. The reversible structure implies

also the existence of such a 1-parameter family of periodic orbits, compare [46, Theorem 4.3 (iii)], [16, Proposition 7.3].

To formulate the further assumptions the authors exploit Fenichel coordinates near the periodic orbit P .

Lemma 2.1 (Fenichel coordinates in [3], Lemma 2.1). Assume Hypothesis (H2.1), (H2.2) and (H2.4). For any small

neighbourhood of the periodic orbit P there is a smooth change of coordinates x 7→ v such that

(i) v = (vc, vss, vuu) ∈ S1 × I2δ , where Iδ := [−δ, δ], for some constant δ > 0.

(ii) In the v-coordinates system (2.1) possesses the structure

vc = 1 +Ac(v, µ) vssvuu

vss = (−α+Ass(v, µ)) vss

vuu = (α+Auu(v, µ)) vuu

=: f(v, µ). (2.2)

The quantities Ac, Ass, Auu are smooth functions such that Ass(vc, 0, 0, µ) ≡ 0 and Auu(vc, 0, 0, µ) ≡ 0.

(iii) The vector field (2.1) is reversible and the action of the reverser R of Hypothesis (H2.1) is described by the

following map:

R : S1 × Iδ × Iδ → S1 × Iδ × Iδ

(vc, vss, vuu) 7→ R(vc, vss, vuu) := (−vc, vuu, vss).

2.1 Unperturbed snaking – snakes and ladders 15

According to Lemma 2.1 let δ > 0 be a sufficiently small constant and Iδ := [−δ, δ]. The authors introduce the sections

Σin := S1 × vss = δ × Iδ, Σout := S1 × Iδ × vuu = δ,

cf. Figure 2.1. These sections are (reversible) R-images of each other.

vss

vuu

P

δ

δ

Wuuloc (P )

W ssloc(P )

Σin

Σout

Figure 2.1: The cross-sections Σin and Σout together with the stable and unstable manifolds, W ssloc(P ) and Wuu

loc (P )of P . [38, Figure 5]

Next the authors claim the existence of an EtoP cycle. This cycle is determined by the intersection of the stable and

unstable manifolds of E and P . Denote by W s(E, µ) the stable manifold of E and by Wuu(γϕ, µ) the 1-dimensional,

unstable leaves of P with base point γϕ = γ(ϕ, µ). Furthermore, the following set is introduced

Γ :=

(ϕ, µ) ∈ S1 × Jµ |Ws(E, µ) ∩Wuu(γϕ, µ) ∩ Σout 6= ∅

.

The first equation in (2.2) provides that γ(ϕ, µ) = ϕ. It is assumed that Γ is the graph of a function.

Hypothesis (H2.5). The set Γ is the graph of a smooth function z : S1 → Jµ, where Jµ is the interior of Jµ.

Moreover it is assumed that

(i) There exist ℓm, ℓM ∈ S1 such that: z′(ϕ) = 0 if and only if ϕ ∈ ℓm, ℓM and

(ii) z′′(ℓm), z′′(ℓM) 6= 0 and

(iii) zout(ℓm) < zout(ℓM).

Consequently z has precisely two critical points, one minimum taken at ℓm and one maximum taken at ℓM, which are

both nondegenerate. Next the authors state assumptions on the stable and unstable manifolds of E within the section

Σout.

16 2 Preliminaries

Hypothesis (H2.6). There exist an open neighbourhood UΓ of Γ in S1 × Jµ, an ǫ > 0 and a smooth function

gss : UΓ → Iδ such that:

(ϕ, vss, δ) ∈W s(E, µ) ∩ Σout | |vss| < ǫ, (ϕ, µ) ∈ UΓ = (ϕ, gss(ϕ, µ), δ) | (ϕ, µ) ∈ UΓ.

Moreover there exists a constant b > 0 such that |Dµgss(ϕ, µ)| > b > 0 for all (ϕ, µ) ∈ UΓ.

Observe that combining Hypotheses (H2.5) and (H2.6) implies that in UΓ

gss(ϕ, z(ϕ)) ≡ 0 ∀ϕ, gss(ϕ, µ) = 0 if and only if µ = z(ϕ). (2.3)

Furthermore, as a direct consequence of the reversibility one obtains

z(ϕ) = z(−ϕ). (2.4)

In [3] it is explained that nondegenerate minima and maxima of the function z correspond to saddle-node bifurcations

of continuation curves of homoclinic orbits to E, see [3, Figure 1.5, Figure 1.6]. Exploiting Hypothesis (H2.1)–(H2.6)

the authors derive the existence and location of the continuation curve of homoclinic orbits (the two snaking curves).

Theorem 2.1 (Symmetric 1-homoclinic orbits, Theorem 2.2 in [3]). Assume that Hypotheses (H2.1)–(H2.6) are met,

then there are constants L0 ≫ 1 and α > 0 so that the following is true: for each L > L0, (2.1) has a symmetric

homoclinic orbit for µ ∈ Jµ that spends time 2L in the neighbourhood of P if and only if

µ = µ∗(L,ϕc) = z(ϕc + L) +O(

e−2αL)

for a ϕc ∈ 0, π.

Furthermore define Sϕc:= (µ,L) = (µ∗(L,ϕc), L) |L > L0, ϕ = 0, π, then asymmetric homoclinic orbits (that

correspond to the rungs in the bifurcation diagram) are determined by the following theorem.

Theorem 2.2 (Asymmetric 1-homoclinic orbits, Theorem 2.3 in [3]). Assume that Hypotheses (H2.1)–(H2.6) are

met. There are an α > 0, natural numbers n∗ ≫ 1 and smooth curves

asym(·, n, ϕc), asym(·) : [0, 1] −→ R2, 1 ≤ i ≤ k,

defined for n > n∗ and ϕc ∈ 0, π so that the following is true:

(i)

asym(s, n, ϕc) = asym(s) + (0, ϕc + 2πn) +O(e−2αn), s ∈ [0, 1].

The curves asym(s, n, ϕc) begin for s = 0 in Sϕcnear the maximum of z and terminate for s = 1 in Sϕc

near

the minimum of z.

(ii) Equation (2.1) has an asymmetric homoclinic orbit for µ ∈ Jµ that spends time 2L in the neighbourhood of γ(·, µ)

if and only if (µ,L) lies on one of the curves asym(s, n, ϕc) with s 6= 0, 1.

(iii) The start and end points of the curves asym(s, n, ϕc) for s = 0, 1 correspond to pitchfork bifurcations of the

2.1 Unperturbed snaking – snakes and ladders 17

symmetric homoclinic orbits described in Theorem 2.1; no other pitchfork bifurcations occur on Sϕc. Furthermore,

the curves asym(s) are obtained from solutions (L,ϕ) of

z(L+ ϕ) = z(L− ϕ), L ∈ S1,

upon setting µ = z(L+ ϕ).

The proofs of Theorem 2.2 and Theorem 2.2 are based on the existence and estimates of solutions of (2.2) in the

vicinity of the hyperbolic periodic orbit P (restricted to the zero level set of H) that start at some time −L in the

section Σin and hit at the time L the section Σout. The task to find such solutions is called Shilnikov problem.

In the context of unperturbed snaking the authors exploited the following lemma, see [3, Lemma 3.1], that yields the

solution of the Shilnikov problem near a hyperbolic periodic orbit:

Lemma 2.2 (The Shilnikov problem, Lemma 3.1 in [3]). Assume Hypotheses (H2.1), (H2.2), (H2.4). There exist

positive constants L0 and α so that the following is true for all L > L0 and ϕ ∈ S1: there is a unique solution v(x),

also referred to as v(x, ϕ), of (2.2), defined for x ∈ [−L,L] that stays in the vicinity of P , so that

v(−L) ∈ Σin, v(L) ∈ Σout, vc(0) = ϕ.

Furthermore, we have

v(−L) =(

ϕ− L+O(e−2αL), δ, δe−2α(µ)L(1 +O(e−2αL)))

v(L) =(

ϕ+ L+O(e−2αL), δe−2α(µ)L(1 +O(e−2αL)), δ)

v(0) =(

ϕ, δe−α(µ)L(1 +O(e−2αL)), δe−α(µ)L(1 +O(e−2αL)))

.

The solution v(x) is smooth in (ϕ, µ, L), and the error estimates in (2.5) can be differentiated. Furthermore, we have

v(x,−ϕ) = Rv(−x, ϕ), ϕ ∈ S1, |x| ≤ L.

In particular, the solution v(x, ϕ) is R-reversible, with v(0) ∈ Fix(R), if and only if ϕ = 0, π.

A solution of (2.2) is homoclinic to E, if and only if, it lies in W s(E, µ) ∩Wu(E, µ). Since Beck et al. are interested

only in homoclinic solutions near the EtoP cycle, these solutions have to hit Σin and Σout, i.e.

vss(L,ϕ) = gss(ϕ+ L+O(e−2αL), µ),

vuu(−L,ϕ) = gss(ϕ− L+O(e−2αL), µ), ϕ ∈ S1.

Exploiting Hypothesis (H2.6), the fact that RW s(E, µ) = Wu(E, µ) and Lemma 2.2 one derives the determination

equation of homoclinic orbits:

δe−2α(µ)L(1 +O(e−2αL)) = gss(ϕ+ L+O(e−2αL), µ),

δe−2α(µ)L(1 +O(e−2αL)) = gss(ϕ− L+O(e−2αL), µ), ϕ ∈ S1. (2.5)

18 2 Preliminaries

To solve the determination equation one applies (2.3), which yields that

µ = z(ϕ+ L) +O(e−2αL) and µ = z(ϕ− L) +O(e−2αL).

Hence solutions of (2.5) are determined by the zero level set of the function Z0:

Z0(ϕ,L) = z(ϕ+ L)− z(ϕ− L) + e(L,ϕ), where e(L,ϕ) = O(e−2αL). (2.6)

Exploiting (2.4) the authors derive in [3] the complete zero level set of Z0. From the zero level set of Z0 one obtains the

structure of the bifurcation diagram. Recall that in what follows we describe the perturbations of the original snaking

scenario in terms of perturbations of the function Z0 and that, as in [3], the bifurcation diagrams of the perturbed

scenarios correspond to the zero level set of that perturbed function. Therefore the structure of Z0 is also crucial for

our analysis. More precisely we need the following result of [3]:

Lemma 2.3 (The technical results of [3]). Assume Hypotheses (H2.1)–(H2.6).

(i) Z0(L, 0) = Z0(L, π) = 0 and the lines ϕ = 0, ϕ = π correspond to the snaking curves of symmetric homoclinic

orbits in the bifurcation diagram.

(ii) There is a neighbourhood of Z0 = 0 and critical points

Pϕc

i (n) := (Lϕc

i (n), ϕc), i ∈ m,M, n ∈ N, Lϕc

i (n) = ℓi − ϕc + 2πn+O(e−2αn), ϕc ∈ 0, π,

such that

D(L,ϕ)Z0(L,ϕ) = 0 ⇔ ∃n ∈ N : (L,ϕ) = Pϕcm (n) or (L,ϕ) = Pϕc

M (n)

These points correspond in the bifurcation diagram to pitchfork points, where asymmetric homoclinic orbits bi-

furcate from symmetric homoclinic orbits.

(iii) There is an n0 ∈ N such that for L > L0 the zero level set Z0 = 0 consists of unique solution curves. Any of

these curves

(L(·, n), ϕ(·, n)) : [0, 1]→ R× S1

starts end ends in one of the critical points of Z0 = 0, where it intersects with three other solutions curves.

The branches where ϕ(·, n) 6∈ 0, π are related to asymmetric homoclinic orbits and correspond to the vertical

rungs in the bifurcation diagram.

(iv) The points Pϕc

i (n), i ∈ m,M, ϕc ∈ 0, π, n > n0, are saddle points of Z0:

D2(L,ϕ)Z0(L

ϕc

i (n), ϕc) =

(

0 2z′′out(ℓi) +O(

e−2αn)

2z′′out(ℓi) +O(

e−2αn)

0

)

.

To illustrate the situation, the zero level set of Z0 is depicted in the left panel in Figure 1.2, where the right panel in

Figure 1.2 displays the corresponding set (µ∗(ϕ,L), L) |Z0(ϕ,L) = 0 in the bifurcation diagram.

2.2 Fenichel coordinates 19

2.2 Fenichel coordinates

The snaking analysis in [3] is based on the existence of so-called Fenichel coordinates, cf. Lemma 2.1. These are par-

ticular (local) coordinates near the periodic orbit P , which reflect the structure of the vector field f . Such coordinates

were introduced at first by Fenichel in the context of slow/fast systems [21]. Since in our investigations we use the

approach of [3], we need Fenichel coordinates to formulate our setup and to perform our analysis. The authors in [3]

use the conservative character of Equation (2.1) to reduce the analysis and the construction of the Fenichel coordinates

to a 3-dimensional space. Here we consider also a scenario without the presence of a first integral (see Chapter 4).

Consequently we have to operate (in that case) in 4-dimensions. Moreover, the construction of the Fenichel coordinates

is strongly connected with the stable and unstable manifolds of the periodic orbit P . In [3] those manifolds are both

2-dimensional. Perturbing the first integral does also change these dimensions. In fact, in the setup of Chapter 4 the

dimensions of the involved manifolds change their dimensions.

To our knowledge there are no assertions concerning Fenichel coordinates in literature that meets our structural

assumptions. Therefore we carry out the construction of these coordinates. In this section we describe the preliminaries

and the final structure of the vector field in Fenichel coordinates. The construction of the Fenichel coordinates is done

in Chapter 6. Finally we want to remark that the Fenichel coordinates used in [3] are a special case of the Fenichel

coordinates we derive here.

We assume the following setup: Let

x = f(x, µ, λ), x ∈ R4, (µ, λ) ∈ Jµ × Jλ ⊂ R2 (2.7)

be a Ck-smooth family of differential equations, where Jµ, Jλ are intervals with 0 ∈ Jλ.

The following Hypotheses (H2.7)–(H2.9) extend the foregoing Hypotheses (H2.1)–(H2.4) to the vector field (2.7)–

including the perturbation parameter λ.


f(Rx, µ, 0) = −Rf(x, µ, 0), ∀x ∈ R, µ ∈ Jµ.

In other words, the vector field f is reversible, if λ = 0.

Hypothesis (H2.8). For λ = 0 Equation (2.7) possesses a smooth first integral H : R4× Jµ → R, H = H(x, µ) with

H(Rx, µ) = H(x, µ).

Hypothesis (H2.9). For all µ ∈ Jµ and all λ ∈ Jλ Equation (2.7) possesses a periodic orbit P = γt := γ(t, µ, λ)

with minimal period 2π, which satisfies the following:

(i) The family γt depends smoothly on µ, λ.

(ii) γ(0, µ, 0) ∈ Fix(R).

(iii) H(γ(t, µ, 0), µ) = 0 and DxH(γ(t, µ, 0), µ) 6= 0 for all t ∈ R, µ ∈ Jµ.

(iv) Beside the Floquet multiplier 1, the periodic orbit γt possesses the Floquet multipliers

e2παss

< e2παe

< e2παuu

and e2παss

< 1 < e2παuu

, ∀µ ∈ Jµ, λ ∈ Jλ,

20 2 Preliminaries

ακ = ακ(µ, λ), κ = ss, e, uu. Further there is an αe = αe(µ, 0) > 0 such that αe = λαe, ∀ (µ, λ) ∈ Jµ × Jλ.

We want to note explicitly that by Hypothesis (H2.9) (iv)

αe(µ, λ) = 0 ⇔ λ = 0.

Since the existence of a first integral implies the existence of two Floquet multipliers 1, the above line yields that for

λ 6= 0 Equation (2.7) possesses no first integral.

Theorem 2.3 (Fenichel coordinates). Consider Equation (2.7), assume Hypotheses (H2.7)–(H2.9) and let Jλ be

sufficiently small. For any small neighbourhood of the periodic orbit P there is a smooth change of coordinates and a

rescaling of time (x, t) 7→ (v, t) such that

(i) v = (vc, ve, vss, vuu) ∈ S1 × I3δ , where Iδ := [−δ, δ], for some constant δ > 0.

(ii) There are constant c, C > 0 such that c t ≤ t ≤ C t and if x ∈ P then t = t.

(iii) In the v-coordinates System (2.7) possesses the structure

vc = 1 +Ac(v, µ, λ)vssvuu

ve = λ[

(αe +Ae(vc, ve, µ, λ) ve) ve +B(v, µ, λ) vss vuu]

vss = (αss +Ass(v, µ, λ)) vss

vuu = (αuu +Auu(v, µ, λ)) vuu

=: f(v, µ, λ). (2.8)

The quantities Ac, Ae, Ass, Auu, B are smooth functions and Ass/uu(vc, 0, 0, 0, µ, λ) ≡ 0.

(iv) Any level set of the first integral H is given by ve = c, for some constant c ∈ R. In particular the zero level

set of H is given by:

H−1(0) = ve = 0.

(v) If the perturbation parameter λ = 0, then the vector field (2.8) restricted to the level set ve = 0 is reversible

and the action of the reverser R, given by Hypothesis (H2.7), is described by the following map:

R : S1 × 0 × Iδ × Iδ → S1 × 0 × Iδ × Iδ

(vc, 0, vss, vuu) 7→ R(vc, 0, vss, vuu) := (−vc, 0, vuu, vss). (2.9)

Moreover, for any vss, vuu ∈ Iδ it holds true that R(vc, ve, vss, vuu) | vc ∈ S1, ve ∈ Iδ = (vc, ve, vuu, vss) | vc ∈

S1, ve ∈ Iδ.

As mentioned, Fenichel coordinates have already been used in the context of snaking analysis, see e.g. [3, 35, 37, 38].

The main difference in the present situation is that the Floquet exponent αe → 0, when λ tends to zero. That means

that at the critical value λ = 0, there is a change of the dimension of the centre manifold of P .

The proof of Theorem 2.3 is somewhat technical and we postpone it to Chapter 6.

2.3 The solution of the Shilnikov problem 21

2.3 The solution of the Shilnikov problem

As indicated in Sections 2.1 solutions of a Shilnikov problem near the periodic orbit P are essential for the analysis

in [3]. Lemma 2.2 yields this solution for a 3-dimensional system near a hyperbolic periodic with 1-dimensional stable

and unstable manifolds. This solution suffices for [3], since due to the existence of a first integral, the analysis is

reduced a 3-dimensional subspace (a level set of the first integral). In the context of this thesis (see Chapter 4) we are

in a more general setup, where no first integral is present. Therefore Lemma 2.2 is not applicable in our situation.

There are several works that address the solution of the Shilnikov problem. As an example we refer to the work of

Schecter, who solves the Shilnikov problem near a normally hyperbolic invariant manifold [67] and in [68] in more

general framework. In the setting of slow-fast systems the Shilnikov problem is investigated by Krupa, Sandstede and

Szmolyan [44]. In contrast to the work of Schecter, the authors derive not only estimates of the solutions, but extract

also the leading order terms. In this thesis we derive a solution of the Shilnikov problem that is based on the work of

Krupa, Sandstede and Szmolyan.

We consider an ODE system with a structure as given in Theorem 2.3 (iii). Further we assume Hypotheses (H2.9).

That means we consider a system of differential equations with structure of (2.8) and this ODE possesses a periodic

orbit with the Floquet exponents

αss(µ, λ) < αe(µ, λ) ≤ 0 < αuu(µ, λ).

Further let ν > 0 be some arbitrarily small constant and define

αss := (1− ν)αss(µ, λ), αuu := (1− ν)αuu(µ, λ), αe(λ) := (1− ν)αe(µ, λ),

α(µ, λ) := min−αss(µ, λ), αuu(µ, λ), := α− supµ∈Jµ,λ∈Jλ

|αe(µ, λ)| .

Theorem 2.4. Assume Hypothesis (H2.7)-(H2.9). There exists positive constants L0, δ0, λ0 such that for any L >

L0 > 0, ϕ ∈ S1, δ0 > δ > 0, λ ∈ Jλ, |λ| < λ0 and all µ ∈ Jµ the boundary value problem

v = f(v, µ, λ), where vss(−L) = δ, vc(0) = ϕ, ve(−L) = λ ξe, vuu(L) = δ

has a unique smooth solution v(·) = v(·, µ, λ) = (vc(·, µ, λ), ve(·, µ, λ), vss(·, µ, λ), vuu(·, µ, λ)) on [−L,L]. Moreover,

the following estimates hold:

v(−L) =(

ϕ− L, λ ξe, δ, ∆uue−2αuuL)

+(

O(

e−(αuu+α)L)

, 0, 0, λO(

e−(2αuu−αe)L)

+O(

e−(2αuu+α)L)

)

,

v(L) =(

L+ ϕ, λ ξe∆ee2αeL, ∆sse2α

ssL, δ)

+(

O(

e(αss−α)L

)

, λ2O(

e(αe+2αe)L

)

+ λO(

e−L)

, λO(

e(2αss+αe)L

)

+O(

e(2αss−α)L

)

0)

,

where ∆ss = ∆ss(L, δ, ϕ, ξe, µ, λ), ∆e = ∆e(L,ϕ, ξe, µ, λ), ∆uu = ∆uu(L, δ, ϕ, ξe, µ, λ) are some smooth, positive,

bounded functions. Moreover ∆ss = ∆ss0 (ϕ, ξe, δ, µ, λ) + λ

[

∆sse (ϕ, ξe, µ, λ) + O

(

eαeL)]

, for some smooth, bounded

non-zero functions ∆ss0 ,∆

sse . Furthermore, if λ = 0, then ∆uu is independent of L and ∆e ≡ 1. The solution v(·) is

22 2 Preliminaries

smooth in z = L,ϕ, ξe, µ and the following estimates hold true

Dz v(−L, ) =( d

dz(ϕ− L) +O

(

e−2αL)

,d

dz(λ ξe), 0, O

(

e−2αuuL)

)

,

Dz v(L) =( d

dz(ϕ+ L) +O

(

e−2αL)

, λO(

e2αeL)

, O(

e2αssL)

, 0)

.

If z = λ, then the same estimates as above hold true, for vc, vss, vuu. The ve component satisfies the following

estimates. If λ 6= 0, then Dλ ve(L) = d

dλ (λ ξe∆ee2α

eL) + λO(

e3αeL)

+ O(

e−L)

and if λ = 0, then the (one-side)

derivative Dλ ve(L) = ξe +O

(

e−αL)

.

Moreover, estimates for higher derivatives of v(·) can be obtained by differentiating the estimates of the first derivatives.

We give the proof of Theorem 2.4 in Chapter 7. There we adapt the analysis applied in the work by Krupa, Sandstede

and Szmolyan [44, Theorem 4, Section 5], where a similar Theorem in the context of slow/fast waves is shown.

23

Chapter 3

Nonreversible perturbations

In this chapter we investigate nonreversible perturbations of the snaking scenario described in [3] or Section 2.1,

respectively. More precisely, we consider a class of perturbations that destroy the reversible symmetry of the vector

field (2.1), but keep its conservative structure. As in [3] we are interested in the continuation of homoclinic orbits

near an EtoP cycle, connecting an equilibrium E with a periodic orbit P . In accordance with [3], we assume that P

is hyperbolic within the zero level set of the first integral, cf. Section 2.1 Hypothesis (H2.4).

In the unperturbed system the continuation curves of homoclinic orbits from a snakes-and-ladders structure, as it

is shown in [3]. But under a symmetry breaking perturbation the pitchfork bifurcations of the snakes-and-ladders

structure unfold, which alters the overall bifurcation diagram. We show that typically two different scenarios may

occur: stacked isolas or two intertwining criss-cross curves.

The chapter possesses the following structure. In Section 3.1 we formulate the precise setup and the final result, see

Theorem 3.1. In Section 3.2 we give the proof of Theorem 3.1. Finally we discuss briefly a geometric explanation

of our analytical results by means of a Poincare map. Moreover we address the problem of explaining a homotopy

between the continuation scenarios of isolas and criss-cross curves. Both discussions are done in Section 3.3.

Finally we want to remark that the results of this section are already published in [38].

3.1 Setup

We consider a Ck-smooth family of differential equations

x = f(x, µ, λ), x ∈ R4, (µ, λ) ∈ Jµ × Jλ ⊂ R2, (3.1)

where Jµ, Jλ are compact intervals with 0 ∈ Jλ. We denote by ϕt(·)t∈R the flow of (3.1). We assume the following:


f(Rx, µ, 0) = −Rf(x, µ, 0), ∀x ∈ R4, µ ∈ Jµ.

Hypothesis (H3.2). For λ = 0 Equation (3.1) possesses a smooth first integral H : R4× Jµ → R, with H = H(x, µ)

that respects the reversible structure H(Rx, µ) = H(x, µ).

Hypothesis (H3.3). E := x = 0 is a hyperbolic equilibrium of (2.1) for all (µ, λ) ∈ Jµ×Jλ. Further H(E, µ) = 0.

We denote by W s(E, µ, λ) and Wu(E, µ, λ) the stable and unstable manifolds of E, respectively. Hypotheses (H3.1)

and (H3.3) imply that the equilibrium E belongs to the fixed space Fix(R) = x ∈ R4 |R(x) = x). Hence the dimen-

24 3 Nonreversible perturbations

sions of the manifolds W s(E, µ, λ) and Wu(E, µ, λ) coincide. Consequently dimW s(E, µ, λ) = dimWu(E, µ, λ) = 2.

Furthermore Fix(R) is half the space dimension, dimFix(R) = 2.




(ii) γ(0, µ, 0) ∈ Fix(R).

(iii) H(γ(t, µ, λ), µ) = 0 and DxH(γ(t, µ, λ), µ) 6= 0 for all t ∈ R, (µ, λ) ∈ Jµ × Jλ.

(iv) Beside the two Floquet multipliers 1, the periodic orbit γt possesses the Floquet multipliers

e2παss

, e2παuu

with e2παss

< 1 < e2παuu

, ∀µ ∈ Jµ, λ ∈ Jλ,

where αk = αk(µ, λ), k = ss, uu.

Observe that assertions of Hypotheses (H3.1), (H3.2), (H3.4) satisfy the assumptions of Hypotheses (H2.7)–(H2.9).

This shows that the preliminaries of Theorem 2.3 are satisfied. Hence near the periodic orbit there exists Fenichel

coordinates. According to Hypothesis (H3.2) the first integral H is preserved under the perturbation caused by λ. Of

course any heteroclinic EtoP and PtoE connection, as well as, any homoclinic solution to E has to lie in the same

level set as the equilibrium E, which is H = 0. Consequently we may restrict our considerations to H = 0. We

introduce Fenichel coordinates v = (vc, vss, vuu) in a neighbourhood of P restricted to H = 0. Note that H = 0

is characterised by ve = 0, cf. Theorem 2.3.

According to Hypothesis (H3.4)(ii) the periodic orbit P = γ(t, µ, 0) is symmetric. This implies that γ(π, µ, 0) ∈ Fix(R)

and that the Floquet exponents satisfy αss(µ, 0) = −αuu(µ, 0). Further Hypothesis (H3.4)(iii) implies that (restricted

to H = 0) the orbit P is hyperbolic with 2-dimensional stable and unstable manifoldsW ssloc(P, µ, λ) andW

uuloc (P, µ, λ),

respectively. Moreover, due to Hypothesis (H3.4)(iii) and the compactness of Jµ × Jλ the Floquet exponents are

bounded away from zero. Recall that vc ∈ S1. Hence Equation (2.9) yields that the fixed space Fix(R) of R is given

by

Fix(R) ∩ H = 0 = (0, vss, vss) ∪ (π, vss, vss).

Now let δ > 0 be a sufficiently small constant and Iδ := [−δ, δ]. We define sections

Σin := S1 × vss = δ × Iδ, Σout := S1 × Iδ × vuu = δ,

cf. Figure 2.1. Note that these sections are R-images of each other. Next we define the EtoP cycle, which lies in the

intersection of the stable and unstable manifolds of E and P . Denote by W ss(γϕ, µ, λ) and Wuu(γϕ, µ, λ) the stable

and unstable leaves of P with base point γϕ = γ(ϕ, µ, λ). For λ = 0 those leaves are 1-dimensional manifolds. We

define

Γinλ :=

(ϕ, µ) ∈ S1 × Jµ |Wu(E, µ, λ) ∩W ss(γϕ, µ, λ) ∩ Σin 6= ∅

,

Γout :=

(ϕ, µ) ∈ S1 × Jµ |Ws(E, µ, λ) ∩Wuu(γϕ, µ, λ) ∩ Σout 6= ∅

,

cf. [3] and [37]. As our notation already suggests, we assume:

3.1 Setup 25

Hypothesis (H3.5). Γout is independent of λ.

Due to the first equation in (2.8) the vc-coordinate of γϕ = γ(ϕ, µ, λ) is equal to ϕ. Furthermore we observe that in

the reversible system, i.e. λ = 0, the sets Γin0 and Γout are R-images of each other. We realise the symmetry breaking

by the assumption that Γinλ and Γout are no longer R-images of each other, if λ 6= 0, cf. Hypotheses (H3.7)–(H3.9)

below.

As in [3], we assume that both Γinλ and Γout are the graph of a smooth function.

Hypothesis (H3.6). The sets Γinλ and Γout are graphs of smooth functions zin(·, λ) : S1 → Jµ and zout : S

1 → Jµ,

respectively, where λ ∈ Jλ, and Jµ is the interior of Jµ. Moreover we assume:

(i) The function zin depends smoothly on λ.

(ii) There exist ℓm, ℓM ∈ S1 such that: z′out(ϕ) = 0, if and only if, ϕ ∈ ℓm, ℓM.

(iii) z′′out(ℓm), z′′out(ℓM) 6= 0, zout(ℓm) < zout(ℓM).

In other words, zout has precisely two critical points, one minimum taken at ℓm and one maximum taken at ℓM,

which are both nondegenerate. Recall that nondegenerate minima and maxima of the function zout correspond to

saddle-node bifurcations of the snaking curves in the bifurcation diagram, see [3, Figure 1.5, Figure 1.6]. Note that we

assume that zout does not depend on the perturbation parameter λ. This restriction is not necessary for our analysis

and is only used to simplify our notation.

As a direct consequence of the reversibility at λ = 0, we find that

zin(ϕ, 0) = zout(−ϕ). (3.2)

Next we state assumptions on the stable and unstable manifolds of E:

Hypothesis (H3.7). There exist an open neighbourhood UΓout of Γout in S1 × Jµ, an ǫ > 0 and a smooth function

gss : UΓout × Jλ → Iδ such that for all λ ∈ Jλ

(ϕ, vss, δ) ∈W s(E, µ, λ) ∩ Σout | |vss| < ǫ, (ϕ, µ) ∈ UΓout = (ϕ, gss(ϕ, µ, λ), δ) | (ϕ, µ) ∈ UΓout.

Moreover there exists a constant b > 0 such that |Dµgss(ϕ, µ, λ)| > b > 0 for all (ϕ, µ, λ).

µ0

µvs

ϕ

Γout

graph gs

Σout

UΓout

Figure 3.1: The set Γout within its open neighbourhood UΓout . The graph of gss is drawn for µ = µ0. The dashedlines indicate a possible continuation of the gss outside of UΓout . [38, Figure 6]


Hypothesis (H3.8). There exist an open set UΓin in S1× Jµ, which is for all λ ∈ Jλ an open neighbourhood of Γinλ ,

an ǫ > 0 and a smooth function huu : UΓin × Jλ → Iδ such that for all λ ∈ Jλ

(ϕ, δ, vuu) ∈Wu(E, µ, λ) ∩ Σin | |vuu| < ǫ, (ϕ, µ) ∈ UΓin = (ϕ, δ, huu(ϕ, µ, λ)) | (ϕ, µ) ∈ UΓin.

The reversibility implies

huu(ϕ, µ, 0) = gss(−ϕ, µ, 0).

Therefore, due to Hypothesis (H3.7)

|Dµhuu(ϕ, µ, λ)| > b > 0, (3.3)

for all (ϕ, µ, λ) with |λ| sufficiently small.

Note that the zeros of gss do not depend on λ, which corresponds to the assumption that Γout is independent of λ.

The Hypotheses (H3.6), (H3.7) or (H3.6), (H3.8) imply that

gss(ϕ, µ, λ) = 0 ⇔ µ = zout(ϕ) and huu(ϕ, µ, λ) = 0 ⇔ µ = zin(ϕ, λ), (3.4)

respectively.

Recall that it is proved in [3] that due to Hypotheses (H3.3)–(H3.7) Homoclinic Snaking as depicted in Figure 1.1

occurs in the unperturbed system with λ = 0, cf. Section 2.1. Moreover, recall that homoclinic orbits to E are

determined by the equation (2.6),

Z0(ϕ,L) = z(ϕ+ L)− z(ϕ− L) + e(L,ϕ) = 0, e(L,ϕ) = O(

e−2αL)

.

In order to define the nonreversible perturbations we consider the difference zin(−ϕ, λ)− zout(ϕ). Exploiting (3.2) we

define the smooth function z(ϕ, λ) by:

λ z(ϕ, λ) := zin(−ϕ, λ)− zout(ϕ). (3.5)

The vector field f is nonreversible for λ 6= 0, if z(ϕ, λ) 6≡ 0. We distinguish the following two cases:

Hypothesis (H3.9). z(ℓM , 0) 6= 0, z(ℓm, 0) 6= 0 and sgn[z(ℓM , 0)] = sgn[z(ℓm, 0)].

Hypothesis (H3.10). z(ℓM , 0) 6= 0, z(ℓm, 0) 6= 0 and sgn[z(ℓM , 0)] 6= sgn[z(ℓm, 0)].

Theorem 3.1 (Main theorem for perturbations of the reversibility). Assume Hypotheses (H3.1)–(H3.8) and either

(H3.9) or (H3.10). There exist positive constants L0 and λ0, and there are functions µ∗λ : R+ × S1 → Jµ such that

for L > L0, |λ| < λ0 Equation (2.1) has a homoclinic orbit that spends the time 2L between the cross-sections Σin

and Σout, if (µ,L) lies on a continuation curve(

µ∗λ(Lλ(s), ϕλ(s)), Lλ(s)

)

distinguished by the following: The mapping

(L,ϕ) 7→ (µ∗0(L,ϕ), L) maps the zero level set of Z0 onto the original snakes-and-ladders structure. Further, if λ 6= 0,

and

(i) Hypothesis (H3.9) is met: There is an n0 > 0 such that for all n ∈ N with n > n0 there are closed curves

(Lλ(s), ϕλ(s)) = (Ljλ,n(s), ϕjλ,n(s)), j ∈ 0, π. These curves are close to the boundaries ∂Q0

+(n) and ∂Qπ+(n) or

∂Q0−(n) and ∂Q

π−(n), respectively, cf. Figure 1.2 and Figure 1.4, panel (a).

3.2 Continuation analysis 27

(ii) Hypothesis (H3.10) is met: There are two curves (Lλ(s), ϕλ(s)) = (Lλ,i(s), ϕλ,i(s)), i = 1, 2. Both curves are

close to the zero level set of Z0. Further, the Lλ,i are unbounded. Approaching a saddle point of Z0 these curves

turn alternately to the left or to the right, cf. Figure 1.4, panel (b).

In either instance the union of these curves converges in the Hausdorff distance to Z0 = 0, as λ tends to zero.

If the curves(

L0/πλ,n (s), ϕ

0/πλ,n (s)

)

approach a saddle point of Z0 they turn always in the same direction.

The corresponding continuation curves(

µ∗λ(L

0/πλ,n (s), ϕ

0/πλ,n (s)), L

0/πλ,n (s)

)

, n > n0 or(

µ∗λ(Lλ,i(s), ϕλ,i(s)), Lλ,i(s)

)

, i =

1, 2, respectively are shown on the right of each panel of Figure 1.4. If Hypothesis (H3.9) is met, the continuation

curves form isolas for λ 6= 0. If, on the other hand Hypothesis (H3.10) is met, the homoclinic orbits can be continued

along criss-cross snaking curves.

3.2 Continuation analysis

This section is devoted the proof of Theorem 3.1. As a first step we derive the determination equation for homoclinic

orbits to E in a vicinity of the primary EtoP cycle. For this we assume Hypotheses (H3.1)–(H3.8). Afterwards we

discuss the determination equation under the additional Hypothesis (H3.9) or (H3.10), respectively.

3.2.1 The determination equation for 1-homoclinic orbits

A 1-homoclinic orbit (which we simply address as homoclinic orbit) to E can be conceived as built of three pieces: an

orbit segment in Wu(E) running from E to Σin, a solution connecting Σin and Σout and an orbit segment in W s(E)

running from Σout to E. Let v(·, µ, λ) be a solution of (3.1) in Fenichel coordinates (cf. Theorem 2.3) starting in

Σin and arriving after time 2L in Σout. Then v(·, µ, λ) belongs to a homoclinic orbit to E, if the following coupling

conditions are fulfilled

v(−L, µ, λ) ∈ Σin ∩Wu(E, µ, λ) and v(L, µ, λ) ∈ Σout ∩W s(E, µ, λ),

for some L≫ 1. Note that due to Hypothesis (H3.4) the Floquet exponents αss = αss(µ, λ), αuu = αuu(µ, λ) depend

smoothly on µ and λ. According to Theorem 2.4 we find homoclinic orbits to E by solving

vss(L,ϕ, µ, λ) = gss(vc(L,ϕ, µ, λ), µ, λ),

vuu(−L,ϕ, µ, λ) = huu(vc(−L,ϕ, µ, λ), µ, λ),(3.6)

where ϕ = vc(0, ϕ, µ, λ). Exploiting Theorem 2.4 these equations can be written as

∆sse2αssL +O

(

e(2αss−α)L

)

= gss(

L+ ϕ+O(

e(αss−α)L

)

, µ, λ)

. (3.7)

∆uue−2αuuL +O(

e−(2αuu+α)L)

= huu(

− L+ ϕ+O(

e−(αuu+α)L)

, µ, λ)

, (3.8)

where we recall that αss, αuu depend on µ, λ, what we suppress in the notation. We solve the Equations (3.8) and

(3.7) for

µ = zout(L+ ϕ) + eout(L,ϕ, λ),

µ = zin(−L+ ϕ, λ) + ein(L,ϕ, λ),(3.9)


respectively, where ein(L,ϕ, 0) = O(

e−2αL)

and eout(L,ϕ, 0) = O(

e−2αL)

. We solve exemplarily (3.7), where we

mimic the analysis in [3, Section 4 and Section 5.2]. We exploit the Implicit Function Theorem. To this end we

consider the function

F (L,ϕ, λ, eout, τ) := τ[

∆sse2αssL +O

(

e(2αss−α)L

)

]− gss(

L+ ϕ+ τ O(

e(αss−α)L

)

, zout(L+ ϕ) + eout, λ)

.

Equation (3.4) implies that gss(ϕ, zout(ϕ), λ) ≡ 0. Hence τ = 0, eout = 0 is an initial solution to F = 0. According

to Hypothesis (H3.7), DµF (L,ϕ, µ, λ, 0, 0) 6= 0. Hence there exists a solution eout(L,ϕ, λ, τ) of (3.7). According to

the Implicit Function Theorem, see Theorem 3.2 at the end of this section, and Theorem 2.4 this solution extends to

τ = 1, since there is an L0 such that

∣

∣1− [DeoutF (L,ϕ, λ, 0, 0)]−1 ·Deout

F (L,ϕ, λ, eout, τ)∣

∣ ≤1

2, ∀L > L0, ∀eout ∈ K[0; ǫ], ∀τ ∈ [0, 1], λ ∈ Jλ,

∣

∣[DeoutF (L,ϕ, λ, 0, 0)]−1 · F (L,ϕ, λ, 0, τ)

∣

∣ ≤ǫ

4, ∀L > L0, ∀τ ∈ [0, 1], λ ∈ Jλ.

Thus we define e∗out(L,ϕ, λ) := eout(L,ϕ, λ, 1) and µ∗out(L,ϕ, λ) := zout(L+ϕ)+e

∗out(L,ϕ, λ). Since F (L,ϕ, λ, e

∗out, 1) ≡

0, a Taylor expansion of F with respect to e∗out shows that e∗out = O(

e−2αL)

. Similarly we obtain µ∗in(L,ϕ, λ) :=

zin(−L+ ϕ, λ) + e∗in(L,ϕ, λ). Further, the reversibility implies that µ∗out(L,ϕ, 0) = µ∗

in(L,−ϕ, 0), and hence

e∗out(L,ϕ, 0) = e∗in(L,−ϕ, 0). (3.10)

Lemma 3.1. e∗in(L,ϕ, λ) = e∗in(L,ϕ, 0) + λO(

e−2αL)

and e∗out(L,ϕ, λ) = e∗out(L,ϕ, 0) + λO(

e−2αL)

, uniformly in λ.

We give the proof of this lemma at the end of this section.

Summarising, a homoclinic solution to E exists at the parameter value µ, if both (3.7) and (3.8) are satisfied. Thus

the determination equation for homoclinic orbits to E reads:

Z(L,ϕ, λ) := µ∗out(L,ϕ, λ)− µ

∗in(L,ϕ, λ) = 0.

Using the representation (3.9) together with (3.5), Lemma 3.1 and (3.10), the function Z can be written as

Z(L,ϕ, λ) =zout(L+ ϕ)− zout(L− ϕ) + e∗out(L,ϕ, 0)− e∗out(L,−ϕ, 0)− λ[z(L− ϕ, λ) +O

(

e−2αL)

].

With Z0(L,ϕ) = Z(L,ϕ, 0) the determination equation reads

Z(L,ϕ, λ) = Z0(L,ϕ)− λ[

z(L− ϕ, λ) +O(

e−2αL)]

= 0. (3.11)

Recall that Z0 = 0 is the determination equation of homoclinic orbits of the unperturbed system, so that the snaking

behaviour of the unperturbed system is also represented in this equation. In Lemma 2.3 we have summarised the

results of [3] regarding the structure of the zero level set of Z0. Here we recall that Z0(L, 0) = Z0(L, π) = 0 and that

the critical points of Z0 in Z0 = 0 are given by (ϕc, Lϕc

i (n)) with i ∈ m,M and ϕc ∈ 0, π and are related to the

critical points of zout by

Lϕc

i (n) = ℓi − ϕc + 2πn+O(

e−2αn)

, n ∈ N, n≫ 1. (3.12)


Proof of Lemma 3.1. We show that Dλe∗in(L,ϕ, λ) = O

(

e−2αL)

. To this end we consider, cf (3.9),

Dλµ∗in(L,ϕ, λ) = Dλzin(−L+ ϕ, λ) +Dλe

∗in(L,ϕ, λ). (3.13)

Because µ∗in solves the first equation in (3.6) we find for the left-hand side of (3.13)

Dλµ∗in(L,ϕ, λ) =

N(L,ϕ, µ∗in, λ)

M(L,ϕ, µ∗in, λ)

,

where

N(L,ϕ, µ∗in, λ) :=Dλv

uu(−L,ϕ, µ∗in, λ)−Dϕh

uu(vc(−L,ϕ, µ∗in, λ), µ

∗in, λ) ·Dλv

c(−L,ϕ, µ∗in, λ)

−Dλhuu(vc(−L,ϕ, µ∗

in, λ), µ∗in, λ),

M(L,ϕ, µ∗in, λ) :=−Dµv

uu(−L,ϕ, µ∗in, λ) +Dµh

uu(vc(−L,ϕ, µ∗in, λ), µ

∗in, λ)

+Dϕhuu(vc(−L,ϕ, µ∗

in, λ), µ∗in, λ) ·Dµv

c(−L,ϕ, µ∗in, λ).

The estimates given in Theorem 2.4 yield

Dλµ∗in(L,ϕ, λ) =

O(

e−2αL)

−Dλhuu(−L+ ϕ+O

(

e−2αL)

, µ∗in, λ)

Dµhuu(−L+ ϕ+O(

e−2αL)

, µ∗in, λ) +O

(

e−2αL) .

Note that due to (3.3) the denominator is bounded away from zero. Note further that for arbitrarily large L the points

(−L+ ϕ, µ∗in(L,ϕ, λ)) belong to a compact subset of UΓin . Hence

Dλµ∗in(L,ϕ, λ) = O

(

e−2αL)

−Dλh

uu(−L+ ϕ, µ∗in, λ)

Dµhuu(−L+ ϕ, µ∗in, λ) +O

(

e−2αL) . (3.14)

Next we consider Dλzin(−L+ ϕ, λ) and recall that huu(−L+ ϕ, zin(−L+ ϕ, λ), λ) ≡ 0, see (3.4). Therefore

Dλzin(−L+ ϕ, λ) = −Dλh

uu(−L+ ϕ, µ∗in − e

∗in(L,ϕ, λ), λ)

Dµhuu(−L+ ϕ, µ∗in − e

∗in(L,ϕ, λ), λ)

.

Using e∗in(L,ϕ, λ) = O(

e−2αL)

we find

Dλzin(−L+ ϕ, λ) = O(

e−2αL)

−Dλh

uu(−L+ ϕ, µ∗in, λ)

Dµhuu(−L+ ϕ, µ∗in, λ) +O

(

e−2αL) . (3.15)

Combining (3.14) and (3.15) yields Dλµ∗in(L,ϕ, λ)−Dλzin(−L+ϕ, λ) = O

(

e−2αL)

. Together with (3.13) this proves

the lemma.

Theorem 3.2 (Implicit Function Theorem). Let X,Y, Z be Banach spaces and G ⊂ X and H ⊂ Y . Denote by K(x; r)

(K[x; r]) the open (closed) ball around x with radius r ∈ R. Let

F : X×Y × R→ Z

(x, y, τ) 7→ F (x, y, τ)

be a Ck-smooth function and let (x0, y0) ∈ G×H be a point such that F (x0, y0, 0) = 0. Assume that F possesses the


bounded inverse [DF ]−1 := [DyF (x0, y0, 0)]−1 and that there exists ν > 0, ǫ > 0 such that the following inequalities

are satisfied

∥

∥I − [DF ]−1 ·DyF (x, y, τ)∥

∥

L(Y )≤

1

2, ∀x ∈ K(x0; ν), ∀y ∈ K[y0; ǫ], ∀τ ∈ [0, 1], (3.16)

∥

∥[DF ]−1 · F (x, y0, τ)∥

∥ ≤ǫ

4, ∀x ∈ K(x0; ν), ∀τ ∈ [0, 1]. (3.17)

Here ‖·‖Y , ‖·‖L(Y ) denote the corresponding norm in Y and the norm in the space of linear mappings from Y into itself,

L(Y ). Furthermore, I is the unity in L(Y ). Then there exists a unique Ck-smooth function f : K(x0; ν) × [0, 1] →

K(y0; ǫ) with f(x0, 0) = y0 such that ∀(x, y, τ) ∈ K(x0; ν)×K(y0, ǫ)× [0, 1] :

F (x, y, τ) = 0 ⇐⇒ f(x, τ) = y.

In particular the function y∗ : K(x0; ν)→ K(y0; ǫ) with y∗(x) := f(x, 1) is the unique solution of F (x, y, 1) = 0.

Considering the proof of the Implicit Function Theorem, the proof of Theorem 3.2 is rather immediate, one merely

has to show additionally that the τ range has the stated global character, see for example [26].

3.2.2 Continuation in the perturbed system

We solve the determination equation (3.11) near the zero level set of Z0. This level set consists of points Pϕc

i (n),

Pϕc

i (n) = (Lϕc

i (n), ϕc), i ∈ m,M, ϕc ∈ 0, π,

and curve segments connecting these points as described in Lemma 2.3.

Using the Implicit Function Theorem we show that (3.11) can be (uniquely) solved near compact parts of the line

segments. These parts can be extended into small neighbourhoods of the adjacent points Pϕc

i (n).

Therefore, a small neighbourhood of Pϕc

i (n) contains four points Pϕc

i (n; j;λ), j ∈ l, r, b, t, belonging to four mutual

disjoint curves gained by applying the Implicit Function Theorem, cf. Figure 3.2. Which of these points can be

connected by solution sets of (3.11) depends on the Hypotheses (H3.9) and (H3.10). By means of these hypotheses

we define “barriers”, which separate solution sets of (3.11). Figure 3.2 shows the alignment of those barriers, which

are related to Hypothesis (H3.9), panel (a) or Hypothesis (H3.10), panel (b), respectively, together with the complete

solution sets of (3.11).

In order to verify that the corresponding points Pϕc

i (n; j;λ), j ∈ l, r, b, t can really be connected we use arguments

based on the Morse Lemma.

Continuation near curve segments

We use the Implicit Function Theorem to solve the determination equation along the curve segments of the zero level

set of Z0. Let (L, ϕ) ∈ Z0 = 0, (L, ϕ) 6= Pϕc

i (n). We write L = L+ l and ϕ = ϕ+ ψ and consider

Z(l, ψ, λ) := Z(L+ l, ϕ+ ψ, λ) = 0.

Note that Z(0, 0, 0) = Z0(L, ϕ) = 0 and that D(l,ψ)Z(0, 0, 0) = D(L,ϕ)Z0(L, ϕ) 6= 0, according to Lemma 2.3. This

shows that Z = 0 can be solved by the Implicit Function Theorem near (L, ϕ, 0) either for l = l∗(ψ, λ) or for


ϕπ 2π0

L

π 2π0

L

ϕ

(a) (b)

PπM(n; b;

λ)

PπM (n; l;λ)

PπM (n; t;λ)

PπM (n; r;λ)

PπM (n; t;λ)

PπM (n; r;λ)

PπM (n; l;λ)

PπM(n; b;

λ)

Figure 3.2: Barriers together with the corresponding solution sets of (3.11). [38, Figure 7]

ψ = ψ∗(l, λ). Note that these functions also depend on (L, ϕ), but we suppress this in the notation. Moreover, there

is a λ1 > 0 such that all those l∗ or ψ∗ are defined for |λ| < λ1. This is obvious if L is taken from a compact interval,

i.e. if (L, ϕ) belongs to a single line segment of Z0 = 0. However, due to the structure of D(l,ψ)Z(ℓ, ψ, λ), it remains

also true for arbitrarily large L: According to (3.11), Z(ℓ, ψ, λ) − Z0(L, ϕ) and D(l,ψ)Z(ℓ, ψ, λ) −D(L,ϕ)Z0(L, ϕ) are

uniformly bounded in L for all |λ| < λ1. Combining this with the version of the Implicit Function Theorem 3.2 shows

the existence of l∗, ψ∗ for all |λ| < λ1.

Varying (L, ϕ, 0) along a curve segment between neighbours Pϕ1

c

i1(n1) and P

ϕ2c

i2(n2), ϕ

1c , ϕ

2c ∈ 0, π, n1, n2 ∈ N shows

that there is a smooth solution curve of Z = 0 connecting arbitrarily small neighbourhoods of Pϕ1

c

i1(n1) and P

ϕ2c

i2(n2).

Definition of barriers

We show that there are certain line segments attached to the critical points of Z0 in its zero level set, along which

there are no solution of the determination equation Z = 0.

Let i ∈ m,M and ϕc ∈ 0, π. Recall that Z0(Lϕc

i , ϕc) = 0. Hence, in accordance with (3.11) and (3.12) we find

Z(Lϕc

i (n), ϕc, λ) = −λ(

z(ℓi, λ) +O(

e−2αn))

. (3.18)

Therefore, for sufficiently large n

sgn[

Z(Lϕc

i (n), ϕc, λ)]

= −sgn[λ] · sgn[z(ℓi, λ)]. (3.19)

Let z(ℓi, 0) > 0, and let λ < 0. Then Z(Lϕc

i (n), ϕc, λ) > 0, and according to (3.11) also Z(L,ϕ, λ) > 0 for (L,ϕ) ∈ Q0/π+

adjacent to Pϕc

i (n), (L,ϕ) close to Pϕc

i (n). In this case we define barriers Bϕi (n), i ∈ m,M on which Z is definitely

different from zero by

Bϕcm (n) := Pϕc

m (n) + s(1, 1), Bϕc

M (n) := Pϕc

M (n) + s(−1, 1), |s| small.


Plugging these definitions into Z yields

Z(Bϕcm (n), λ) = zout(ℓm + 2s)− zout(ℓm) +O

(

e−2αn)

− λ(

z(ℓm, λ) +O(

e−2αn))

,

Z(Bϕc

M (n), λ) = zout(ℓM )− zout(ℓM − 2s) +O(

e−2αn)

− λ(

z(ℓM − 2s, λ) +O(

e−2αn))

.

Hence Z(Bϕcm (n), λ) is positive for all s and sufficiently large n and Z(Bϕc

M (n), λ) is positive at least for some small |s|.

The described situation is depicted in Figure 3.3. In a similar way we proceed for λ z(ℓi, 0) > 0.

ϕ

π 2π0

L

B0M (n)

B0m(n)

Bπm(n+ 1)

L0M(n)

Lπm(n)

L0m(n)

LπM(n)

L0M(n+ 1)

Q0−(n)

Q0+(n) Qπ

−(n)

Q0+(n+ 1)

Qπ+(n)

Qπ−(n+ 1)

Figure 3.3: Bifurcation diagram with the (purple) barrier lines B0/πi (n) in the vicinity of the critical points of Z0 in

Z0 = 0. The diagram is drawn for the situation given by Hypothesis (H3.9), more precisely for λ sgn[z(ℓi, 0)] <0, i = m,M . Along the purple lines the function Z is strictly positive. Hence no homoclinic solution exists along thebarrier lines. [38, Figure 8]

Joining of different solution branches of Z = 0

So far we have determined solution branches of Z = 0 connecting small neighbourhoods of adjacent saddle points of

Z0. Here we discuss how near such a saddle point Pϕc

i (n) appropriate branches can be joined. In doing so we exploit

that Z(·, ·, λ) has a saddle point Pϕc

i (n;λ) close to Pϕc

i (n). This saddle point does not belong to the zero level set of Z.

However, Z(Pϕc

i (n;λ), λ) tends to zero as λ tends to zero. Consequently, near Pϕc

i (n;λ) there are two curve segments

belonging to the zero level set of Z. Both curves connect two of the solution branches near the curves segments that

we have derived above.

In order to make this procedure rigorous we observe at first that by Lemma 2.3 and Hypothesis (H3.6) the Hessian

D2(L,ϕ)Z0(L

ϕc

i (n), ϕc) is nondegenerate. Therefore, near Pϕc

i (n) the equation D(L,ϕ)Z(L,ϕ, λ) = 0 can be solved for

(L,ϕ) = (L,ϕ)(λ) =: Pϕc

i (n;λ). It follows immediately that D2(L,ϕ)Z(P

ϕc

i (n;λ), λ) is nondegenerate.

By similar arguments as in paragraph“Continuation near curve segments” there is a λ2 < λ1 such that for i ∈ m,M,

ϕc ∈ 0, π and all n > n0 the points Pϕc

i (n;λ) are defined for all |λ| < λ2.

Lemma 3.2. Z(Pϕc

i (n;λ), λ) 6= 0, for λ 6= 0.


Proof of Lemma 3.2. First we note that Z(Pϕc

i (n; 0), 0) = Z0(Pϕc

i (n)) = 0. Further, due to Lemma 2.3 and (3.18)

we findd

dλZ(Pϕc

i (n;λ), λ)

∣

∣

∣

∣

λ=0

= −z(ℓi, 0) +O(

e−2αn)

.

Due to Hypotheses (H3.9) or (H3.10), respectively, this derivative is different from zero.

However,

limλ→0

Z(Pϕc

i (n;λ), λ) = Z0(Pϕc

i (n)) = 0.

Due to the Morse Lemma, see [24] and Theorem 3.3 at the end of this section, there is an r > 0 such that on the ball

K(Pϕc

i (n;λ), r) centred at Pϕc

i (n;λ) with radius r, there is a smooth coordinate transformation (ϕ,L) 7→ (x, y) such

that

Z(ϕ,L, λ) = Z(Pϕc

i (n;λ), λ)− x2 + y2, where (x, y) = (x, y)(L,ϕ).

Indeed, inspecting the proof of the Morse Lemma we find that there is a λ0 < λ2 such that r can be chosen independently

from i ∈ m,M, ϕ ∈ 0, π, n > n0 and |λ| < λ0.

Note that within K(Pϕc

i (n;λ), r) all level sets of Z, except the one related to Pϕc

i (n;λ), consist of a pair of curves

each running from boundary to boundary of K(Pϕc

i (n;λ), r).

Now choose |λ| sufficiently small, such that the points Pϕc

i (n; j;λ), j ∈ l, r, b, t are in K(Pϕc

i (n;λ), r). Then, each

two of these points are connected by a smooth curve belonging to the zero level set of Z. Which of these points are

connected in this way depends on the alignment of the barriers defined in Section 3.2.2.

Isolas:

Assume Hypothesis (H3.9), and more specifically that z(ℓM , 0) > 0 and z(ℓm, 0) > 0. If λ < 0, the barriers are aligned

as depicted in Figure 3.3. The corresponding continuation curves form isolas. These are depicted in Figure 3.2, panel

(a). Roughly speaking these isolas surround regions Qi−(n). If λ has the opposite sign those isolas surround regions

Qi+(n). As λ tends to zero the isolas tend to Z = 0 in the Hausdorff distance.

Criss-cross snaking:

Assume Hypothesis (H3.10), and more specifically that z(ℓM , 0) < 0, z(ℓm, 0) > 0, and assume λ < 0. For the points

Pϕcm (n) we just repeat the discussion following (3.19). It turns out that the corresponding barriers take the form

Bϕcm (n) := Pϕc

m (n) + s(1, 1). For the barriers at Pϕc

M (n) a similar discussion yields the same alignment, Bϕc

M (n) :=

Pϕc

M (n) + s(1, 1). This situation together with the corresponding criss-cross snaking curves is depicted in Figure 3.2,

panel (b). If λ changes sign the criss-cross snaking curves become descending. As λ tends to zero the criss-cross

snaking curves tend to Z = 0 in the Hausdorff distance.

Theorem 3.3 (Morse Lemma). Let F : R2 × R → R with F = F (L,ϕ, λ) be a smooth function, with critical

points (p(λ), λ), all not degenerated. Furthermore, assume that D2xF (p(0), 0) possesses one negative and one positive

eigenvalue. Then locally around (p(0), 0) there exists smooth change of coordinates (L,ϕ, λ) 7→ (x, y, λ) such that

(p(0), 0) 7→ (p(0), 0) and within these coordinates the function F is given by

F (p(λ), λ)− x2 + y2.

The proof of the Morse Lemma can be found e.g. in [51].


3.3 Discussion

3.3.1 Connection with Poincare maps

On a geometric level the snaking curves of symmetric homoclinic orbits can be explained by means of an appropriate

Poincare map, see [3, 72]. Choosing the Poincare section in the zero level set of the first integral H allows to some

extent to reduce the dynamics of (2.1) to a 2-dimensional map. This is illustrated in Figure 3.4 and in the lower part

of panel (a) in Figure 3.5. Symmetric homoclinic orbits are given by intersections of W s(E) with Wu(E) that lie in

Fix(R) (bullets), whereas other intersections of W s(E) and Wu(E) (squares) correspond to asymmetric homoclinic

orbits. By varying the parameter µ the manifolds W s(E) and Wu(E) will be moved, which gives rise to saddle

nodes of symmetric homoclinic orbits and to a pitchfork bifurcation of asymmetric homoclinic orbits off the branch of

symmetric homoclinic orbits. To explain one snaking curve at full length larger parts of the corresponding manifolds

have to be taken into consideration. For more detailed explanations of that point we refer to [3, 72]. Altogether we

want to emphasise that those considerations only explain the existence of one (of the two) snaking curves of symmetric

homoclinic orbits and the local pitchfork bifurcations.

µ

‖u‖22

Fix(R)E

Wu(E)

W s(E)

Fix(R)

Wu(E)

W s(E)

E

Fix(R)

Fix(R)

E

W s(E)

Wu(E)

E

Wu(E)

W s(E)

P

P

P

P

Figure 3.4: Homoclinic orbits lie within the sections of the stable and unstable manifolds of E near the periodic orbitP . This is illustrated in the right upper drawing that sketch a global Poincare section. The black dots in these panelscorrespond to a symmetric homoclinic orbit, since it intersects with Fix(R). The black squares represent asymmetrichomoclinic orbits. When varying the parameter µ the manifolds W s(E) and Wu(E) begin to move. In this processthe symmetric homoclinic orbits vanish at saddle nodes, see the left upper drawing. The asymmetric homoclinicorbits disappear in fold bifurcations. The lower panels sketches how the heteroclinic connections between E and Pgoverns the bifurcation diagram. The figure is adapted from [3, Figure 1.3]

Breaking the reversibility effects that the symmetry between W s(E) and Wu(E) gets lost. This is illustrated in

the lower parts Figure 3.5, panels (b) and (c). The lack of symmetry in these drawings explains that in these cases

homoclinic orbits merge in folds (and no longer pitchfork bifurcations do occur). Moreover, Figure 3.5, panels (b) and

(c) in illustrates the effect of the Hypotheses (H3.9) and (H3.10) on the relative position of the stable and unstable

3.3 Discussion 35

manifolds of the equilibrium E.

W s(P )

Wu(P )

Wu(E)

W s(E)

Fix(R)

ϕ

µ

zout(ϕ)

(a)

Wu(E)

W s(E)

ϕ

µ

zout(ϕ)

(b)zin(−ϕ, λ)

Wu(E)

W s(E)

ϕ

µ

zout(ϕ)

(c)zin(−ϕ, λ)

P P P

Wu(P )

W s(P ) W s(P )

Wu(P )

Figure 3.5: The qualitative relation between the relative position of zin and zout and the resulting shape of theperturbed manifold Wu(E) is illustrated. Panel (a) displays the unperturbed situation, i.e. λ = 0: The graphsof zout(ϕ) and zin(−ϕ, 0) coincide, and the stable and unstable manifolds of E are R-images of each other. Theupper drawing in panel (b) reflects, for some λ 6= 0, a situation assumed by Hypothesis (H3.9); The lower drawingillustrates the corresponding perturbation of the unstable manifold of the equilibrium – the dashed line represents theunperturbed manifold. In the same way the drawings in panel (c) illustrate the effect of Hypothesis (H3.10). [38, Figure9]

However, by means of these pictures neither the existence of isolas nor the existence of criss-cross snaking can be

deduced, since these phenomenon also involve the second snaking curve of symmetric homoclinic orbits, see Figure 1.4.

3.3.2 Transition from isolas to criss-cross snaking

In this Chapter we have studied symmetry-breaking perturbations of Homoclinic Snaking scenarios. Adapting the

approach in [3, 35] we have shown that the continuation curves of homoclinic orbits in the perturbed system are

generically either isolas or criss-cross snakes.

It is an interesting question if and how the two different bifurcation scenarios can be transformed into one another.

In order to address this problem we consider a homotopy between the two examples considered in the introduction,

which is given by

uxxxx = µu− u− 2uxx + 2u2 − u3 + λ(

τ(

3uxu2xx + u2xuxxx

)

+ (1− τ) (3uxuxx))

, (3.20)

where as above λ = 0.3. If τ = 1, then homoclinic solutions to the equilibrium E = 0 lie on isolas in the bifurcation

diagram, while for τ = 0 the system exhibits criss-cross snaking, compare with Figure 1.5.

Numerical studies presented in [38] show that isolas and criss-cross snaking persist for τ ≈ 1 and τ ≈ 0, respectively,

and that a transition occurs for τ ∈ [0.1, 0.2]. This transition process is illustrated in Figure 3.6.


−0.50 −0.45 −0.40 −0.35

0.10

0.15

0.20

0.25

−0.50 −0.45 −0.40 −0.35

0.10

0.15

0.20

0.25

−0.50 −0.45 −0.40 −0.35

0.10

0.15

0.20

0.25

µ µµ

(a) (b) (c)

Figure 3.6: Break-up of criss-cross snaking into isolas in Equation (3.20). The dark bold curves show large isolas,while the lighter curves are separate isolas of figure-eight type. The values of the homotopy parameter are τ = 0.109in (a), τ = 0.11 in (b) and τ = 0.15 in (c). [38, Figure 10]

If we decrease the parameter from τ = 0.2, then figure-eight type isolas at the lower end of the bifurcation diagram, i.e.

those corresponding to solutions with a narrow middle part, merge to form a larger isola. This occurs in a transcritical

bifurcation near the fold on the left, as indicated in the panels containing magnified parts of panel (c) in Figure 3.6.

The process continues if τ is decreased further, see panels (a) and (b). The values of the homotopy parameter τ at

which further figure-eight isolas merge with the large isola at the bottom of the bifurcation diagram seem to converge

to a finite value, such that at τ = 0.1 the whole criss-cross snaking structure appears to be recovered. (The numerical

results are necessarily incomplete, since only a finite part of the whole structure can be computed.)

These results are clearly different from the generic behaviour established in our analysis. Indeed, since the homotopy

connects the two different scenarios in Hypotheses (H3.9) and (H3.10), there must be a value τ0 such that z(ℓM , λ) = 0

or z(ℓm, λ) = 0. The numerical results suggest that τ0 ≈ 0.1.

37

Chapter 4

General perturbations

In this chapter we analyse a further class of perturbations of the original snakes-and-ladders scenario in [3]. In contrast

to the situation in Chapter 3 the perturbation under consideration destroys not only the reversible symmetry of the

vector field in (2.1), but also the first integral. The loss of the conservative structure changes the dynamics near the

periodic orbit, which causes several structural differences to the system considered in [3] or Chapter 3, respectively.

In fact, there arise some difficulties to overcome.

Due to the loss of the first integral, the considerations can no longer be restricted to a lower dimensional submanifold

(such as the level set). Therefore we need Fenichel coordinates in the entire state space R4 with all the additional

requirements stated in the introduction, see also Chapter 6. Correspondingly we adapt the analysis of Shilnikov

problem, Chapter 7. However, the immediate effect regarding the continuation is that a second continuation parameter

is required. This is due to the fact that in the perturbed system homoclinic orbits are (at least) of codimension-1. In

our analysis the continuation will be processed in the original snaking parameter µ and the perturbation parameter λ.

Observe that this is pretty similar to the numerical continuation in [7], where also an “all structure destroying”

perturbation is considered. Eliminating the perturbation parameter from the bifurcation diagram by a projection the

resulting continuation curves, presented in Figures 4.3 and Figure 4.2, look as the ones in Chapter 3, cf. Figure 1.4.

More precisely the (projected) bifurcation diagram may have two possible structures: stacked isolas or two intertwining

criss-cross curves.

The chapter is organised as follows. In Section 4.1 we formulate the setup and the main continuation result, see

Theorem 4.1. Section 4.2 is devoted to the proof of Theorem 4.1. Finally we compare our analytical results with the

numerical studies of [7] in Section 4.3.

4.1 Setup and main result

We consider a Ck-smooth family of differential equations

x = f(x, µ, λ), x ∈ R4, (µ, λ) ∈ Jµ × Jλ ⊂ R2, (4.1)

where Jµ, Jλ are compact intervals with 0 ∈ Jλ. Again we denote by ϕt(·)t∈R the flow of (4.1). We assume the

following:


f(Rx, µ, 0) = −Rf(x, µ, 0), ∀x ∈ R4, µ ∈ Jµ.

38 4 General perturbations

Hypothesis (H4.2). For λ = 0 Equation (4.1) has a smooth first integral H : R4 × Jµ → R, with H = H(x, µ) that

respects the reversible structure: H(Rx, µ) = H(x, µ).

Hypothesis (H4.3). E := x = 0 is a hyperbolic equilibrium of (4.1) for all (µ, λ) ∈ Jµ × Jλ. Furthermore,

H(E, µ) = 0.

Again we denote by W s(E, µ, λ) and Wu(E, µ, λ) the stable and unstable manifolds of E, respectively. According to

Hypotheses (H4.1) the vector field f is reversible if λ = 0. As for Hypothesis (H3.1) in Chapter 3 this implies that

dimW s(E, µ, λ) = dimWu(E, µ, λ) = 2. Furthermore, Fix(R) = x ∈ R4|R(x) = x is 2-dimensional, dimFix(R) = 2.




(ii) γ(0, µ, 0) ∈ Fix(R).

(iii) H(γ(t, µ, 0), µ) = 0 and DxH(γ(t, µ, 0), µ) 6= 0 for all t ∈ R, µ ∈ Jµ.

(iv) Beside the Floquet multiplier 1, the periodic orbit γt possesses the Floquet multipliers

e2παss

< e2παe

< e2παuu

and e2παss

< 1 < e2παuu

, ∀µ ∈ Jµ, λ ∈ Jλ,

where ακ = ακ(µ, λ), κ = ss, e, uu. Further there is an αe = αe(µ, 0) > 0 such that αe = λαe, ∀ (µ, λ) ∈ Jµ×Jλ.

According to the above Hypotheses the vector field f is neither reversible, nor it possesses a first integral if λ 6= 0. In

what follows we consider the case where λ ≤ 0, which means that

αe = λ αe ≤ 0.

Next we exploit Theorem 2.3 to transform (4.1) near P into Fenichel coordinates. As before we define sections

Σin := S1 × Iδ × vss = δ × Iδ, Σout := S1 × Iδ × Iδ × v

uu = δ.

According to Theorem 2.3 (v), these sections are R-images of each other, RΣin = Σout, if λ = 0.

In what follows we characterise the behaviour of the heteroclinic EtoP and PtoE connections. By definition the EtoP

(PtoE) connection has to lie within the intersection of the unstable (stable) and stable (unstable) manifolds of the

equilibrium E and the periodic orbit P . To describe the intersection of these manifolds more precisely we introduce

the following 2-dimensional, flow invariant manifolds (in Fenichel coordinates)

W ss,e(ϕ, µ, λ) :=

vc = ϕ, vuu = 0

⊂W ss,eloc (P ), W e,uu(ϕ, µ, λ) :=

vc = ϕ, vss = 0

⊂W e,uuloc (P ).

If αe < 0, then W ss,e(ϕ, µ, λ) denotes the stable leaf of P with base point γϕ ∈ P . In this case W ss,e(ϕ, µ, λ) corre-

sponds to the α-conventionally stable manifold [65, Definition 5.1] with α ∈ (αe, αuu). The manifold W e,uu(ϕ, µ, λ)

might be considered (for αe < 0) as a kind of extended unstable leaf of P , but has not a corresponding conventionally

unstable manifold according to [65, Definition 5.1].

4.1 Setup and main result 39

Similarly to Chapter 3 we define sets:

Γinλ :=

(ϕ, µ) ∈ S1 × Jµ |Wu(E, µ, λ) ∩W ss,e(ϕ, µ, λ) ∩ Σin 6= ∅

,

Γout :=

(ϕ, µ) ∈ S1 × Jµ |Ws(E, µ, λ) ∩W e,uu(ϕ, µ, λ) ∩ Σout 6= ∅

.

Hypothesis (H4.5). The set Γout is independent of λ.

As in Chapter 3 this hypothesis is not necessary for the analysis that follows, but simplifies the notation. If λ = 0 the

system is reversible and Γin0 and Γout are R-images of each other.

Again, cf. Hypothesis (H2.5), we assume that Γinλ and Γout are graphs of smooth functions:

Hypothesis (H4.6). The sets Γinλ and Γout are graphs of smooth functions zin(·, λ) : S1 → Jµ and zout(·) : S

1 → Jµ,

respectively, where Jµ is the interior of Jµ and λ ∈ Jλ. Moreover we assume

(i) The function zin depends smoothly on λ.

(ii) There exist ℓm, ℓM ∈ S1 such that: z′out(ϕ) = 0 if and only if ϕ ∈ ℓm, ℓM.

(iii) z′′

out(ℓm) 6= 0, z′′

out(ℓM ) 6= 0, zout(ℓm) < zout(ℓM).

According to Hypothesis (H4.6) the function zout possesses exactly one minimum and one maximum at ℓm and ℓm.

By (iii) both extremes are nondegenerate. Due to the reversibility of f we observe that if λ = 0, then

zin(−ϕ, 0) = zout(ϕ).

The next hypothesis makes assumptions about the intersections of the stable and unstable manifolds of E with Σin

and Σout, respectively.

Hypothesis (H4.7). There exists an open set UΓin in S1 × Jµ, which is for all λ ∈ Jλ an open neighbourhood of

Γinλ , an ǫ > 0, and smooth functions he, huu : UΓin × Jλ → Iδ such that for all λ ∈ Jλ

(ϕ, ve, δ, vuu) ∈Wu(E, µ, λ) ∩ Σin | |ve| < ǫ, |vuu| < ǫ, (ϕ, µ) ∈ UΓin

=

(ϕ, he(ϕ, µ, λ), δ, huu(ϕ, µ, λ)) | (ϕ, µ) ∈ UΓin

.

Hypothesis (H4.8). There exists an open neighbourhood UΓout of Γout in S1 × Jµ, an ǫ > 0 and smooth functions

gss, ge : UΓout × Jλ → Iδ such that for all λ ∈ Jλ

(ϕ, ve, vss, δ) ∈W s(E, µ, λ) ∩ Σout | |ve| < ǫ, |vss| < ǫ, (ϕ, µ) ∈ UΓout

=

(ϕ, ge(ϕ, µ, λ), gss(ϕ, µ, λ), δ) | (ϕ, µ) ∈ UΓout

.

As a direct consequence of Hypothesis (H4.6), Hypothesis (H4.7) and Hypothesis (H4.8) we find that

(

ϕ, ge(ϕ, µ, λ), gss(ϕ, µ, λ), δ)

∈W e,uuloc (P ) ∩ Σout ⇔ gss(ϕ, µ, λ) = 0,

(

ϕ, he(ϕ, µ, λ), δ, huu(ϕ, µ, λ))

∈W ss,eloc (P ) ∩ Σin ⇔ huu(ϕ, µ, λ) = 0.


If UΓin , UΓout are chosen appropriately, then by Hypothesis (H4.6) it follows

gss(ϕ, µ, λ) = 0 ⇔ µ = zout(ϕ) and huu(ϕ, µ, λ) = 0 ⇔ µ = zin(ϕ, λ). (4.2)

Consider the case where λ = 0. Then, as we discussed it in Chapter 3, the equilibrium E lies within H = 0. This

implies that the stable and unstable manifolds of E belong to H = 0 and consequently

ge(ϕ, µ, 0) = 0, he(ϕ, µ, 0) = 0.

The next hypothesis gives some technical assumptions we will need:

Hypothesis (H4.9). (i) There exists a constant b > 0 such that |Dµgss(ϕ, µ, λ)| > b for all (ϕ, µ, λ) ∈ S1×Jµ×Jλ.

(ii) There exists a function ge : UΓout × Jλ → Iδ with ge(ϕ, µ, λ) 6= 0 such that ge(ϕ, µ, λ) = −λ2 ge(ϕ, µ, λ).

(iii) There exists a function he : UΓin × Jλ → Iδ with he(ϕ, µ, λ) 6= 0 such that he(ϕ, µ, λ) = λ he(ϕ, µ, λ).

(iv) sgn[he(ϕ, µ, λ)] = sgn[ge(ϕ, µ, λ)].

Let us briefly discuss the meaning of Hypothesis (H4.9). Hypothesis (H4.9) (i) is related to the solvability condition

that was assumed in [3] and in Hypothesis (H3.7). According to (2.9), the functions gss, huu are related at λ = 0 by

huu(ϕ, µ, 0) = gss(−ϕ, µ, 0). Hence for sufficiently small |λ|

|Dµhuu(ϕ, µ, λ)| > b, ∀ϕ ∈ S1, ∀µ ∈ Jµ. (4.3)

For the sake of simplicity we assume that Jλ is chosen that small so that (4.3) is valid for all λ ∈ Jλ.

Hypothesis (H4.9) (ii) implies that the stable manifold of E does not intersect the (strong) unstable manifold of P ,

if λ 6= 0. Recall that dim(W ss(E)) = dim(Wuuloc (P )) = 2. Hence, in R4, these manifolds do generically not intersect.

Hypothesis (H4.9) assumes (amongst others) this generic situation. Hypothesis (H4.9) (iii) has a similar effect on the

manifolds Wu(E) and W ssloc(P ). Comparing Hypothesis (H4.9) (ii) and (iii) we see that within the sections Σin/out

the trace of the manifolds Wuu(E) and W ss(E) move at different speed away from zero as λ moves off zero.

Hypothesis (H4.9) (iv) is a solvability condition, since any solution starting with positive (negative) ve in Σin, hits

Σout at a positive (negative) value of ve, cf. Theorem 2.4. Consequently ve cannot change its sign.

Finally we introduce the following notation

zin(−ϕ, λ)− zout(ϕ) =: λ z(ϕ, λ) (4.4)

and assume that

Hypothesis (H4.10). z(ℓm, λ) 6= 0, z(ℓM , λ) 6= 0, ∀λ ∈ Jλ.

Moreover, we assume one of the following hypotheses

Hypothesis (H4.11). sgn[z(ℓm, 0)] = sgn[z(ℓM , 0)].

Hypothesis (H4.12). sgn[z(ℓm, 0)] = −sgn[z(ℓM , 0)].


Theorem 4.1 (Main theorem for perturbations of the reversibility and the first integral).

Assume Hypotheses (H4.1)–(H4.10) and either (H4.11) or (H4.12). There exists a positive constant L0 and there

are functions (µ∗,k, λ∗,k) : R+ × S1 → Jµ × Jλ, k = 0, 1, such that for all L > L0 Equation (4.1) has a homo-

clinic orbit that spends the time 2L between the cross-sections Σin and Σout, if (µ, λ, L) lies on a continuation curve(

µ∗,k(L(s), ϕ(s)), λ∗,k(L(s), ϕ(s)), L(s))

distinguished by the following:

λ∗,0(L,ϕ) ≡ 0, λ∗,1(L,ϕ) < 0, limL→∞

λ∗,1(L,ϕ) = 0,

and the mapping (L,ϕ) 7→ (µ∗,0(L,ϕ), L) maps the zero level set of Z0 onto the original snakes-and-ladders structure.

Let k = 1. Then, if

(i) Hypothesis (H4.11) is met: There is an n0 ∈ N such that for all n ∈ N with n > n0 there closed curves

(L(s), ϕ(s)) = (Ljn(s), ϕjn(s)), s ∈ S

1, j ∈ 0, π. These curves are close to the boundaries ∂Q0+(n) and ∂Q

π+(n)

or ∂Q0−(n) and ∂Q

π−(n), respectively, cf. Figure 3.2, panel (a) and Figure 3.3.

Further, limn→∞ dist(

(Ljn(s), ϕjn(s)), Z0 = 0

)

= 0 for all s ∈ S1, j ∈ 0, π.

(ii) Hypothesis (H4.12) is met: There are two curves (L(s), ϕ(s)) = (Li(s), ϕi(s)), s ∈ R+, i = 1, 2. Both curves are

close to the zero level set of Z0. Further, the Li are unbounded. Approaching a saddle point of Z0 these curves

turn alternately to the left or to the right, cf. Figure 3.2, panel (b).

Further, lims→∞ dist(

(Li(s), ϕi(s)), Z0 = 0)

= 0, i = 1, 2.

4.2 Continuation analysis

In this section we prove Theorem 4.1. First of all we derive the determination equations for homoclinic orbits to

the equilibrium E in the vicinity of the primary heteroclinic EtoP cycle. By exploiting Theorem 2.4 we solve these

equations piecewise locally near the solution of the unperturbed determination equation. After that we investigate the

way the local solutions connect to global solutions.

4.2.1 The determination equation for 1-homoclinic orbits

As in Section 3.2.1 we consider 1-homoclinic orbits (recall that we address them just as homoclinic orbits) to be

composed of three parts: two pieces running between the equilibrium E and the cross-sections Σin and Σout, as well

as, the middle part that spends the time 2L between these cross-sections. Consequently, under the assumptions of

Hypotheses (H4.1)–(H4.4) any homoclinic solution of (4.1) corresponds to a solution v(·, µ, λ) of (2.8) that satisfies

v(−L, µ, λ) ∈ Σin ∩Wu(E, µ, λ) and v(L, µ, λ) ∈ Σout ∩W s(E, µ, λ). (4.5)

Assuming Hypothesis (H4.5)–(H4.8) condition (4.5) yields the following determination equations for homoclinic orbits

ve(−L,ϕ, ξe, µ, λ) = he(vc(−L,ϕ, ξe, µ, λ), µ, λ),

vuu(−L,ϕ, ξe, µ, λ) = huu(vc(−L,ϕ, ξe, µ, λ), µ, λ),

vc(0, ϕ, ξe, µ, λ) = ϕ,

ve(L,ϕ, ξe, µ, λ) = ge(vc(L,ϕ, ξe, µ, λ), µ, λ),

vss(L,ϕ, ξe, µ, λ) = gss(vc(L,ϕ, ξe, µ, λ), µ, λ).

(4.6)


If λ = 0, then our considerations are reduced to the level set H = 0 = ve = 0. Consequently ve(−L,ϕ, ξe, µ, 0) = 0

and ve(L,ϕ, ξe, µ, 0) = 0. The remaining equations reduce to the equation set of the unperturbed system considered

in [3], cf. Section 2.1.

In what follows we assume therefore that λ < 0.

As in Chapter 3 we exploit Theorem 2.4 to detect appropriate solutions v(·, µ, λ) of (4.6). Recall that P possesses the

Floquet exponents

αss(µ, λ) < αe(µ, λ) ≤ 0 < αuu(µ, λ).

Moreover we recall the definitions



|αe(µ, λ)| ,

for some arbitrarily small constant ν > 0. Assume that the interval Jλ is sufficiently small, then by Theorem 2.4 the

determination equation (4.6) reads

λ ξe = λ he(ϕ− L+O(

e−2αL)

, µ, λ), (4.7)

∆uue−2αuuL + λO(

e−(2αuu−αe)L)

+O(

e−(2αuu+α)L)

= huu(ϕ− L+O(

e−2αL)

, µ, λ), (4.8)

λ ξe∆ee2αeL + λ2O

(

e(αe+2αe)L

)

+ λO(

e−L)

= −λ2 ge(L+ ϕ+O(

e−2αL)

, µ, λ), (4.9)

∆sse2αssL + λO

(

e(2αss+αe)L

)

+O(

e(2αss−α)L

)

= gss(L+ ϕ+O(

e−2αL)

, µ, λ), (4.10)

where we simplified the O-terms in the vc estimates and suppressed the µ, λ dependency of the α-exponents. As in

Chapter 3 we write

µ = zin(−L+ ϕ, λ) + ein and µ = zout(L+ ϕ) + eout,

where ein and eout are small quantities. The first representation of µ is related to the coupling in Σin and is exploited

in the corresponding Equations (4.7), (4.8). Correspondingly the second representation of µ is related to the coupling

in Σout and used in the Equations (4.9), (4.10).

We solve at first (4.7) and (4.9). Plugging in (4.7) into (4.9) and dividing by λ we obtain

λ = F (L,ϕ, ein, eout, λ) :=1

−λ ge(. . .)ve(L, µ, λ) = −

1

ge(. . .)

(

(

he(. . .) ·∆ee2αeL + λO

(

e(αe+2αe)L

)

+O(

e−L)

)

,

where

he(. . .) = he(ϕ− L+O(

e−2αL)

, zin(ϕ− L, λ) + ein, λ),

ge(. . .) = ge(L+ ϕ+O(

e−2αL)

, zout(L+ ϕ) + eout, λ).

Note that due to Hypothesis (H4.9) the mapping F is well-defined. Further, concerning F we observe the following.


There is an L0 such that for all L > L0 and λ ∈ [λ0, 0]:

λ0 < F (L, . . . , λ) < 0, F (L, . . . , λ = λ0) < 0, (4.11)

where Jλ ∩ R−0 =: [λ0, 0]. Calculating the derivative of F w.r.t λ gives

∂

∂λF (L, . . . , λ) =

[

Dλve(L, µ, λ) +Dξev

e(L, µ, λ) dd λ (λ he(. . .))

]

(−λ ge(. . .))− ve(L, µ, λ) dd λ (−λ g

e(. . .))

λ2 (ge(. . .))2

= −Dλv

e(L, µ, λ)

λ ge(. . .)+O

(

e2αeL)

.

If λ 6= 0, then Dλ ve(L, µ, λ) = d

dλ (λ ξe∆ee2α

eL) + λO(

e3αeL)

+O(

e−L)

= 2λ ξe∆ee2αeL ·L+O

(

e2αeL)

and if λ = 0,

then Dλ ve(L, µ, 0) = ξe + O

(

e−αL)

, by Theorem 2.4. By Hypotheses (H4.9) sgn[ge(. . .)] = sgn[he(. . .)] = sgn[ξe].

Consequently, if L is sufficiently large, the positivity ∆e yields for λ ∈ [λ0, 0) that

∂

∂λF (L, . . . , λ) < 0. (4.12)

Now (4.11) and (4.12) imply immediately that for fixed L > L0 and arbitrary ϕ, ein, eout the mapping F , considered

as a function of λ, has a unique fixed point

λ∗(L,ϕ, ein, eout) ∈ (λ0, 0).

Applying the Implicit Function Theorem reveals that λ∗ is smooth. Further, λ∗(L, . . .) = F (L, . . . , λ∗(L, . . .)) and the

assumption that λ∗(L, . . .) is bounded away from zero, are in contradiction to each other. Hence

λ∗(L, . . .)→ 0, as L→∞.

Considering λ∗(L, . . .) = F (L, . . . , λ∗(L, . . .)) more thoroughly reveals that αeL = λαe L→ −∞, if L tends to ∞ and

furthermore, that for each ν > 0 there is an Lν > 0 such that

|λ∗(L, . . .)| > e−νL, ∀L > Lν . (4.13)

Similarly we find that the derivatives Dz1λ∗ and Dz1z2λ

∗, z1, z2 = L,ϕ, ein, eout, tend to zero as L→∞.

Next we consider (4.8) and (4.10) with λ = λ∗(. . .) and solve these equations for ein and eout, respectively. As in

Chapter 3 we do this by applying the Implicit Function Theorem at ’L =∞’. We define

G(L,ϕ, ein, eout, τ) :=

(

τ · cuu(L,ϕ)− huu(

ϕ− L+ τ · O(

e−2αL)

, zin(−L+ ϕ, λ∗) + ein, λ∗)

τ · css(L,ϕ)− gss(

L+ ϕ+ τ · O(

e−2αL)

, zout(L+ ϕ) + eout, λ∗)

)

,

where cuu(L,ϕ), css(L,ϕ) denote the left hand side of (4.8) and (4.10):

cuu(L,ϕ) := ∆uue−2αuuL + λO(

e−(2αuu−αe)L)

+O(

e−(2αuu+α)L)

,

css(L,ϕ) := ∆sse2αssL + λO

(

e(2αss+αe)L

)

+O(

e(2αss−α)L

)

.


According to (4.2) the function G(L,ϕ, 0, 0, 0) ≡ 0. Exploiting Theorem 2.4 and the estimates on λ∗(. . .) and Dzλ∗(. . .)

yields that D(ein,eout)G(L,ϕ, 0, 0, 0) is invertible (at least for large L). By the Implicit Function Theorem there exist

unique solutions ein(L,ϕ, τ) and eout(L,ϕ, τ) of G = 0. These solutions coincide with the solutions of (4.8) and (4.10),

if τ = 1. We observe that (3.16) and (3.17) are satisfied for sufficiently large L. Hence τ can (indeed) be extended to

1 and we may define e∗in(L,ϕ) := ein(L,ϕ, 1) and e∗out(L,ϕ) := eout(L,ϕ, 1). Further G(L,ϕ, e∗in, e∗out, λ

∗) ≡ 0 and a

Taylor expansion of huu(. . .) and gss(. . .) w.r.t. e∗in and e∗out show that

e∗in(L,ϕ) = O(

e−2αL)

and e∗out(L,ϕ) = O(

e−2αL)

.

Finally differentiating (4.2) w.r.t. L,ϕ, λ and exploiting that ddzG(L,ϕ, e

∗in, e

∗out, λ

∗) ≡ 0 for z = L,ϕ gives

Dze∗in(L,ϕ) = O

(

e−2αL)

and Dze∗out(L,ϕ) = O

(

e−2αL)

.

Next we redefine

λ∗(L,ϕ) := λ∗(L,ϕ, e∗in(L,ϕ), e∗out(L,ϕ))

and write

µin(L,ϕ) = zin(−L+ ϕ, λ∗(L,ϕ)) + e∗in(L,ϕ),

µout(L,ϕ) = zout(L+ ϕ) + e∗out(L,ϕ).

In order to solve system (4.7)–(4.10) conclusively it remains to solve µin(L,ϕ) = µout(L,ϕ), or equivalently the

equation

Z(L,ϕ) := zout(L+ ϕ)− zin(−L+ ϕ, λ∗(L,ϕ)) + e∗out(L,ϕ)− e∗in(L,ϕ) = 0. (4.14)

Recall the definition of Z0 = zout(L+ ϕ)− zout(L− ϕ) +O(

e−2αL)

in (2.6), where in our setting zout corresponds to

the function z in the unperturbed system, cf. (4.4). By the estimates on e∗in(. . .), e∗out(. . .), Equation (4.14) can be

written as a perturbation of Z0 = 0,

Z(L,ϕ) = Z0(L,ϕ)− λ∗ z(L− ϕ, λ∗) +O

(

e−2αL)

= 0. (4.15)

We observe that (4.15) possesses the same structure as (3.11), with the difference that λ∗ is not a free parameter. But

since λ∗(L,ϕ) → 0 for L → ∞, the function Z is ”close to” Z0. To solve (4.15) we introduce the following function

that possesses an artificial parameter:

Z(L,ϕ, τ) := Z0(L,ϕ)− τ ·[

λ∗ z(L− ϕ, λ∗) +O(

e−2αL)]

. (4.16)

Equation Z(L,ϕ, 1) = 0 is equivalent to (4.15) and Z(L,ϕ, 0) = 0 is simply Z0 = 0. Recall that the zero level set of

Z0 is thoroughly investigated in [3] and the main results of that work regarding the function Z0 are summarised in

Lemma 2.3.


4.2.2 Continuation near curve segments

In this section we solve Equation (4.15) in the vicinity of the zero level set of Z0. Recall that according to Lemma 2.3

the zero level set of Z0 consists of solution curves (L(·, n), ϕ(·, n)) : [0, 1]→ R× S1. We show the following lemma.

Lemma 4.1. There is an open covering U of Z0 = 0 and an n0 ∈ N such that for any n ∈ N with n > n0 the

following statements are true.

(i) Any critical point Pϕc

i (n) of Z0 is contained in exactly one set Uϕc

i (n) ∈ U , i ∈ m,M. Within any of these

sets there exist precisely two solution curves corresponding to the equation Z = 0, cf. (4.15).

(ii) For any n > n0 the diameter of the sets Uϕc

i (n) is bounded below by a positive constant.

(iii) There are exactly four sets Uj ∈ U , j ∈ 1, . . . 4 such that Uϕc

i (n)∩Uj 6= ∅. And within any Uj there is a unique

solution curve(

L(·, n) + ℓ∗(·, n), ϕ(·, n) + ψ∗(·, n))

: [0, 1]→ R× S1 solving Z = 0.

Basically the proof of Lemma 4.1 proceeds as the proof of the corresponding assertions in Section 3.2.2. At first we

solve (4.15) in some neighbourhoods (Uj) of Z0 = 0 that are away from the critical points Pϕc

i (n) of Z0, given

by Lemma 2.3. There we may exploit the Implicit Function Theorem to solve Z = 0 either for L = L∗(ϕ, τ) or for

ϕ = ϕ∗(L, τ), which yields a unique solution curve in any of these neighbourhoods. This is done in the next subsection.

After that we solve Z = 0 near the critical points Pϕc

i (n). For that we apply the Morse Lemma. This yields the

neighbourhoods Uϕc

i (n) wherein Z = 0 possesses exactly two solution curves.

Solving Z = 0 away from the critical Points

According to Lemma 2.3, Z0 = 0 consists of unique solution curves (L(·, n), ϕ(·, n)). These curves start and end in

the critical points of Z0. In the following we pick out one of those solutions curves and solve Z = 0 exemplarily near

that curve, but away from the critical points of Z (and Z0, respectively).

Inspecting the definition of Z, we see that (L(·, n), ϕ(·, n)) solves Z(L,ϕ, 0) = 0. Thus we may apply the Implicit

Function Theorem along the initial solution (L(·, n), ϕ(·, n)). This yields a solution of Z(L,ϕ, τ) = 0 for small τ . After

that we show that this solution extend to τ = 1 by means of Theorem 3.2.

To make this procedure more rigorous we decompose (L,ϕ) = (L(s, n) + ℓ, ϕ(s, n) + ψ), s ∈ (0, 1) and define

Z(ℓ, ψ, s, n, τ) := Z(L(s, n) + ℓ, ϕ(s, n) + ψ, τ)

= Z0(L+ ℓ, ϕ+ ψ)− τ ·[

λ∗ z(L+ ℓ− ϕ− ψ, λ∗) +O(

e−2αn)]

,

where we observe that O(

e−2α(L+ℓ))

= O(

e−2αn)

. We see that Z(L,ϕ, τ) = 0 is equivalent to Z(ℓ, ψ, s, n, τ) = 0.

Hence (ℓ, ψ, τ) = (0, 0, 0) is an initial solution for Z = 0. Next we check that the Jacobian of Z is invertible. We infer

from Lemma 2.3 that there are intervals IZ(n) ( (0, 1) such that for all s ∈ IZ(n):

∥

∥D(ℓ,ψ)Z(0, 0, s, n, 0)∥

∥ =∥

∥D(L,ϕ)Z0(L(s, n), ϕ(s, n))∥

∥ ≥ bZ > 0, (4.17)

for some positive constant bZ . Hence the Implicit Function Theorem yields that Z(ℓ, ψ, s, n, τ) = 0 possesses a unique

solution curve (ℓ∗(s, n, τ), ψ∗(s, n, τ)), for all s ∈ IZ(n) and sufficiently small ℓ, ψ. It remains to show that this solution

extend to τ = 1:


Lemma 4.2. There is an n0 ∈ N such that for all n ∈ N with n > n0 the solution (ℓ∗(s, n, τ), ψ∗(s, n, τ)) of Z = 0

exists at τ = 1 for all s ∈ IZ(n).

Proof of Lemma 4.2. Let s ∈ IZ(n) such that |DℓZ(0, 0, s, n, 0)| > bZ . Consequently we can solve Z(ℓ, ψ, s, n, τ) = 0

for ℓ by means of the Implicit Function Theorem. To extend this solution to τ = 1 we have to show (see Theorem 3.2)

that there is an ǫ > 0 and a neighbourhood Uψ(n) of zero such that for all τ ∈ [0, 1]:

∥

∥

∥I −[

DℓZ(0, 0, s, n, 0)]−1

DℓZ(ℓ, ψ, s, n, τ)∥

∥

∥ ≤1

2, ∀ℓ ∈ (−ǫ, ǫ), ∀ψ ∈ Uψ(n) (4.18)

∥

∥

∥

[

DℓZ(0, 0, s, n, 0)]−1

Z(0, ψ, s, n, τ)∥

∥

∥ ≤ǫ

4, ∀ψ ∈ Uψ(n). (4.19)

Inequality (4.19) is immediate since on the one handDℓZ(0, 0, s, n, 0) > bZ and on the other hand Z(0, ψ, s, n, λ, 1)→ 0

for ψ → 0 and n → ∞. The argument to show (4.18) is based on a Taylor expansion of DℓZ and the fact that

λ∗(. . .)→ 0 for n→∞. Thus the solution ℓ∗ extends to τ = 1.

Now let |DℓZ(0, 0, s, n, 0)| ≤ bZ . By the definition of IZ(n) it follows that |DψZ(0, 0, s, n, 0)| > bZ . Thus we can solve

Z(ℓ, ψ, s, n, τ) = 0 for ψ. To extend that solution to τ = 1 we show inequalities analogous to (4.18) and (4.19).

Solving Z = 0 near the critical Points

So far we have solved Z = 0 near the sets (

L(s, n), ϕ(s, n))

|n ∈ N, s ∈ IZ(n). These sets possess a positive

(Hausdorff) distance to the critical points of Z. It remains to solve Z = 0 in the vicinity of that critical points, where

the Implicit Function Theorem is not applicable. There we solve Z = 0 by applying the Morse Lemma (Theorem 3.3),

as it is done in Section 3.2.2 or [3, 38], respectively. At first we show that in the vicinity of any critical point Pϕc

i (n)

of Z0 there are nondegenerate critical points of Z. We exploit the following lemma:

Lemma 4.3. There is an n0 ∈ N such that for all n ∈ N, n > n0 there are small neighbourhoods of the critical points

Pϕc

i (n), i ∈ m,M such that within this neighbourhoods there are uniquely defined points Pϕc

i (n, τ) with

DLZ(Pϕc

i (n, τ), τ) = 0, as well as, DϕZ(Pϕc

i (n, τ), τ) = 0, ∀τ ∈ [0, 1].

Further D2(L,ϕ)Z(P

ϕc

i (n, 1), 1) is invertible and D2(L,ϕ)Z(P

ϕc

i (n, 1), 1) possesses exactly one positive and one negative

eigenvalue.

Proof of Lemma 4.3. The arguments of the proof are similar to those in in Section 3.2.2. The critical points

Pϕc

i (n) of Z0 are nondegenerate. Hence the Implicit Function Theorem yields that there are critical points of the

function Z nearby, if λ∗(. . .) is small. We combine this with the assertion that λ∗(. . .) and its derivatives tend to zero

for large L. This shows the existence of the points Pϕc

i (n, τ) for all τ ∈ [0, 1].

It remains to consider the second derivative of Z. According to Lemma 2.3 (and Hypothesis (H4.10)), the matrix

D2(L,ϕ)Z0(P

ϕc

i (n)) is invertible. The invertible matrices form an open set. Therefore it suffice to show that the matrix

D2(L,ϕ)Z(P

ϕc

i (n, 1), 1) becomes arbitrarily close to D2(L,ϕ)Z0(P

ϕc

i (n, 1)). For that it is enough that

‖Pϕc

i (n, 1)− Pϕc

i (n)‖ → 0, for n→∞. (4.20)

According to the definition of Pϕc

i (n, 1), the definition of the function Z and since λ∗(. . .) → 0, if n → ∞, it follows


that

DLZ0(Pϕc

i (n, 1)) −−−−→n→∞

0, DϕZ0(Pϕc

i (n, 1)) −−−−→n→∞

0.

The uniqueness of the critical points of Z0 implies (4.20).

Next we recall that the matrix D2(L,ϕ)Z0(P

ϕc

i (n)) possesses exactly one positive and one negative eigenvalue, see again

Lemma 2.3. Further recall that the derivatives of λ∗(. . .) tend to zero, if L→∞. Consequently D2(L,ϕ)Z(P

ϕc

i (n, 1), 1)

has a positive and a negative eigenvalue, which is a direct consequence of the continuity of the determinant and the

continuous dependence of the zeros of a quadratic polynomial on the polynomial’s coefficients.

Remark 4.1. Observe that though we derived the critical points Pϕc

i (n, 1) by the Implicit Function Theorem with

Pϕc

i (n) as initial solution, the statements of the Lemma are not immediate. In fact, since Pϕc

i (n, 1) are the critical

points of Z at τ = 1, which might be far away from the critical points Pϕc

i (n) of Z = 0 at τ = 0.

Lemma 4.3 proves that the Morse Lemma (Theorem 3.3) can be applied to the function Z. This yields that there is

a smooth change of coordinates (L,ϕ) 7→ (x, y) such that near Pϕc

i (n, 1):

Z(L,ϕ) = Z(Pϕc

i (n, 1), 1)− (x)2 + (y)2, where (x, y) = (x, y)(L,ϕ).

Solving Z(Pϕc

i (n, 1), 1)− (x)2 + (y)2 = 0 in the (x, y)-coordinates gives the solution

x = ±√

Z(Pϕc

i (n, 1), 1) + (y)2.

If Z(Pϕc

i (n, 1), 1) 6= 0, this yields two disjoint solution curves (locally near Pϕc

i (n, 1) ), whereas if Z(Pϕc

i (n, 1), 1) = 0,

then the two solution curves intersect within the critical point Pϕc

i (n, 1). We prove the following lemma:

Lemma 4.4. There is an n0 ∈ N such that for all n ∈ N with n > n0 and ϕc ∈ 0, π:

Z(Pϕc

i (n, 1), 1) 6= 0

and sgn[

Z(Pϕc

i (n, 1), 1)]

= −sgn[λ∗z(ℓi, λ∗)], i = m,M .

Proof of Lemma 4.4. Recall that for the critical points Pϕc

i (n) = Pϕc

i (n, 0) of Z0 holds Z0(Pϕc

i (n)) =

Z(Pϕc

i (n), 0) = 0. Moreover, see Lemma 4.3, the critical points Pϕc

i (n, 1) are obtained by means of the Implicit

Function Theorem. We consider the derivative of the function Z with respect to τ . By exploiting Lemma 4.3 we find

d

dτZ(Pϕc

i (n, τ), τ) =

(

DLZ(Pϕc

i (n, τ), τ)

DϕZ(Pϕc

i (n, τ), τ)

)

·d

dτPϕc

i (n, τ)− λ∗ · z(Pϕc

i (n, τ), λ∗) +O(e−αn)

= −λ∗ · z(Pϕc

i (n, τ), λ∗) +O(e−αn), ∀τ ∈ [0, 1].

According to (4.13) and since z(ℓi, λ∗) 6= 0, it follows that the O-term decreases faster than λ∗ · z(. . .). Therefore it fol-


lows that ddτZ(P

ϕc

i (n, τ), τ) 6= 0 and sgn[ ddτZ(Pϕc

i (n, τ), τ)] = −sgn[λ∗z(ℓi, λ∗)] for sufficiently large L. Consequently

the function τ 7→ Z(Pϕc

i (n, τ), τ) is only zero at τ = 0 and in particular Z(Pϕc

i (n, 1), 1) 6= 0.

To simplify our notation we define

Pϕc

i (n) := Pϕc

i (n, 1) and Z(L,ϕ, 1) := Z(L,ϕ).

Pπm(n)

Pϕcm (n)

Uπm(n, λh)

U...

U...

U...

U...

Z0 = 0

Figure 4.1: The solution of Z = 0 near the critical point Pπm(n). In green the 4 neighbourhoods Uj , j ∈ 1, . . . , 4

with the (green) solutions of Z = 0 that are obtained by the Implicit Function Theorem (away from the criticalpoints). The black line through the critical point Pπ

m(n) denotes a barrier line, which cannot be crossed by anysolutions. Thus only adjacent solutions w.r.t. the barrier can connect via the purple solution branches that are dueto the Morse Lemma.

Now we finish the proof of Lemma 4.1. To this end let us summarise what we have done so far:

Any of the sets (

L(s, n), ϕ(s, n))

| s ∈ IZ(n) is a subset of the level set Z0 = 0. We have solved Z = 0 in some

neighbourhood Uj of (

L(s, n), ϕ(s, n))

| s ∈ IZ(n) by the Implicit Function Theorem. According to (4.17), the sets

Uj do not contain any of critical points Pϕc

i (n) of Z0. Further the sets Uj depend on the size of bZ in (4.17). If

bZ → 0, then the distance between Uj and the two corresponding critical points Pϕc

i (n), i = m,M gets zero. But for

sufficiently large n (and consequently small |λ∗|) we can choose bZ arbitrarily small without violating (4.18) or (4.19).

Hence the distance between Uj and Pϕc

i (n) becomes as small as convenient.

On the other hand the Morse Lemma yields two disjoint solution curves of Z = 0. These curves exist in some

neighbourhoods Uϕc

i (n) of the critical points Pϕc

i (n) of Z. But the distance between Pϕc

i (n) and Pϕc

i (n), becomes

arbitrarily small for sufficiently large n, see (4.20). Inspecting the Proof of the Morse Lemma we find that the

neighbourhoods Uϕc

i (n) do not shrink unbounded for n→∞.

Consequently Uϕc

i (n) and Uj intersect for large n. Since the solutions obtained by the Implicit Function Theorem and


the Morse Lemma are unique, they ’fit together’. In other words the local solution curves connect to global solutions

curves. This is illustrated in Figure 4.1. The different possibilities of such connections are discussed in the next section.

4.2.3 The bifurcation diagram

In the previous Section we have shown that for sufficiently large n ∈ N, there is a unique solution of Z = 0 in any

of the sets Uj and two solution branches in each of the sets Uϕc

i (n), i = m,M,n ∈ N. We have already pointed out

that whenever the sets Uj and Uϕc

i (n) intersect, then the solutions curves within these sets have to couple. Now we

investigate how this coupling and investigate takes place. As in Section 3.2.2 we define barriers on which the function

Z is certainly different from zero. The barriers ’prevent’ the solutions in two of the sets Uj from coupling with one

of the curves in Uϕc

i (n). According to our considerations in the last section, it follows that this curve has to connect

with the solutions in the two other sets Uj , see again Figure 4.1.

To set up the barriers we investigate the sign of Z along the lines Pϕc

i (n)+(s, s), | s ∈ R and Pϕc

i (n)+(s,−s) | s ∈ R.

Lemma 4.5. There is an n0 ∈ N such that for all n ∈ N with n > n0 holds

sgn[

Z(Pϕc

i (n))]

= −sgn[

λ∗ z(ℓi, 0)]

6= 0, i = m,M.

Proof of Lemma 4.5. We recall the definition of Z, (4.16), and that Z0(Pϕc

i (n)) = 0. Further, λ∗(. . .) → 0, if

n→∞, but with an arbitrarily small exponential rate, see (4.13). Hence

sgn[

Z(Pϕc

i (n))]

= −sgn[

λ∗ z(ℓi, λ∗)]

. (4.21)

By Hypothesis (H4.10), the function z(ℓi, λ) 6= 0 for all λ ∈ Jλ and therefore sgn[

z(ℓi, λ∗)]

= sgn[

z(ℓi, 0)]

.

Lemma 4.6. Let c > 0 be an arbitrarily constant. There is an n0 ∈ N such that for all n ∈ N with n > n0 and

i = m,M :

(i) sgn[

Z(Pϕc

i (n))]

= 1 =⇒ Z(Pϕc

i (n) + (s, s)) > 0, ∀s ∈ (−c, c),

(ii) sgn[

Z(Pϕc

i (n))]

= −1 =⇒ Z(Pϕc

i (n) + (s,−s)) < 0, ∀s ∈ (−c, c).

Proof of Lemma 4.6. We consider at first (i). Let s ∈ R, then

Z(Pϕc

i (n) + (s, s)) = zout(ℓm + 2s+O(e−2αn))− zout(ℓm +O(e−2αn)) +O(e−2αn)

− λ∗(s, n) · z(ℓm +O(e−2αn), λ∗(s, n)), (4.22)

where λ∗(s, n) := λ∗(Pϕc

i (n)+(s, s)). To calculate the sign of Z(Pϕc

i (n)+(s, s)) we recall that ℓm denotes the minima

of zout. Therefore

zout(ℓm + 2s+O(e−2αn))− zout(ℓm +O(e−2αn)) > −|O(e−2αn)|, (4.23)

for all s ∈ R. Further we observe that though λ∗(s, n) → 0, if n → ∞ for all s ∈ (−c, c), there is a n0 ∈ N and an

arbitrarily small constant ν such that for all n > n0,

|λ∗(s, n)| > e−νn, ∀s ∈ (−c, c),


cf. (4.13). Therefore, if n is sufficiently large, then |λ∗(s, n)| > |O(e−2αn)|. Consequently, plugging in (4.23) into

(4.22) and exploiting Lemma 4.5 and (4.13) assertion (i) follows. Assertion (ii) can be shown similarly.

0 π 2π

L

µ

L0m(n)

L0m(n+ 1)

L0M (n)

LπM (n+ 1)

L0M (n+ 1)

Lπm(n)

LπM (n)

Lπm(n+ 1)

L

ϕ

Figure 4.2: The zero level set of Z and the corresponding bifurcation diagram. Depicted is the case where the barriersat the critical points force isolas. [Scenario 1 in Table 4.1]

Depending on the type of barriers at the critical points, the unique solutions given by the Implicit Function Theorem

connect with one of the solution curves given by the Morse Lemma. The argument is the following:

Four of the sets Uj intersect with the neighbourhood Uϕc

i (n), cf. Figure 4.1, Consequently, in any Uj there is a point x

on the solution given by the Implicit Function Theorem and that lies in Uϕc

i (n). According to the uniqueness property,

each x lies on one of the solution curves obtained by the Morse Lemma. We have seen that the barrier exists for n > n0

and that it depends only on the size of n0, but not on n. But the sets Uj tend to Z0 = 0 for growing n. Furthermore

inspecting the Proof of the Morse Lemma, we get that Uϕc

i (n) does not shrink unbounded for growing n. Thus the

barrier separates Uϕc

i (n) into two sets. Since Uj is close to Z0 = 0, exactly two of the points x lie in one of these

sets. Hence only those two points can and therefore have to lie on the same solution curve given by Morse Lemma.

At any critical point Pϕc

i (n) of Z0 there are exactly two scenarios how the solutions curves can connect. More precisely,

either the barrier at Pϕc

i (n) is given by Pϕc

i (n) + (s, s) | s ∈ (−c, c) or by the set Pϕc

i (n) + (s,−s) | s ∈ (−c, c).

Equation (4.21) shows that the sign of Z in the critical points depends only on the sign of z. In particular the sign is

independent of n > n0 and ϕc. Lemma 4.6 yields that (for fixed i = m,M) at each of the critical points Pϕc

i (n) there

exists the same“sort” of barrier. Hence the solution curves connect in the same way at each critical point Pϕc

i (n). This

leads to four scenarios how the solution connect at the ”global scale”. Each scenario yields either isolas or criss-cross

snaking, as it is illustrated in the Figures 4.2 and 4.3. The following Table 4.1 specifies the four scenarios:

4.3 Discussion 51

Scenario sgn[

Z(Pϕc

M (n))]

barrier sgn[

Z(Pϕcm (n))

]

barrier type1 1 Pϕc

M (n) + (s,−s) 1 Pϕcm (n) + (s, s) isolas

2 1 Pϕc

M (n),+(s,−s) -1 Pϕcm (n) + (s,−s) criss-cross

3 -1 Pϕc

M (n),+(s, s) 1 Pϕcm (n) + (s, s) criss-cross

4 -1 Pϕc

M (n),+(s, s) -1 Pϕcm (n) + (s,−s) isolas

Table 4.1: Collocation of the signs of Z at the critical points and the corresponding snaking scenarios.

0 π 2π

L

µ

L0m(n)

L0m(n+ 1)

L0M (n)

LπM (n+ 1)

L0M (n+ 1)

Lπm(n)

LπM (n)

Lπm(n+ 1)

L

ϕ

Figure 4.3: The zero level set of Z and the corresponding bifurcation diagram. Depicted is the case where the barriersat the critical points force criss-cross snaking. [Scenario 3 in Table 4.1]

4.3 Discussion

Recall that the analysis in this chapter was motivated by the numerical considerations of Burke et al. in [7]. Now we

briefly compare this numerical studies with our analytical approach. To this end we sketch at first some of the results

of [7]:

The authors study the cubic Swift-Hohenberg equation (1.1), where they add the additional perturbation term η ∂3xu:

ut = µu− (1 + ∂2x)2u+ η ∂3xu+ 2u2 − u3. (4.24)

Burke et al. are interested in steadily drifting states and use the wave ansatz u(t, x) = v(x − λt) =: v(y) to solve

(4.24). With that they derive the corresponding time-independent ODE:

−λ ∂yv = µv − (1 + ∂2y)2v + η ∂3yv + 2v2 − v3. (4.25)

According to the perturbation term η ∂3yv, this ODE has neither a reversible structure, nor a first integral. Steadily

drifting states in (4.24) correspond to homoclinic solutions of (4.25) to the equilibrium 0. The authors show that for


fixed η these solution may be continued within the snaking parameter µ and the drift speed λ. Furthermore they

derive the (projected) bifurcation diagram, which plots µ versus the L2 norm of the solution v. They find that this

bifurcation diagram consists of stacked isolas, similar to Figure 1.5 panel (b). Furthermore Burke et al. observe that

the size of the isolas depends on the value of the perturbation parameter η. A qualitative sketch of this is given in

Figure 4.4.

‖u‖22

µ

Figure 4.4: Isolas corresponding to homoclinic orbits of Equation (4.25) at different values of the perturbationparameter η. Not shown is the (not constant) drift speed λ. Dashed printed is the primary snakes-and-laddersstructure. Figure adapts qualitatively [7, Figure 4].

Though the analytical investigations in Chapter 4 are inspired by the above example, there are some differences

between the situation in Equation (4.25) and our setup here. First of all we start straightaway in the context of ODEs

and instead of three parameters, our scenario possesses only two parameters, the snaking parameter µ that corresponds

to the snaking parameter µ in (4.25) and the parameter λ. In our setting λ = λ(L) acts as perturbation parameter,

as well as, continuation parameter. Therefore λ(L)→ 0 as L→∞. Consequently the ODE in our analytical scenario

“tends” to the unperturbed system. This distinguishes our setup qualitatively from the scenario of Burke et al. – and

also from the purely nonreversible perturbations considered in Chapter 3, where the perturbation parameter λ remains

constant during the continuation, as well.

Nevertheless our snaking results and the numerical results of Burke et al. display a high degree of similarity. Our

analysis predicts the emergence of isolas (beside criss-cross snaking) in the bifurcation diagram, as it is observed

in [7]. As there, the continuation of homoclinic orbits takes place within two parameters– (µ, λ), where µ = µ(L) and

λ = λ(L). Moreover we observe the effect that the size of the corresponding isolas varies, too. In fact, recall that

the bifurcation diagram is related to the zero level set of the function Z, which is a perturbation of the function Z0,

corresponding to the classical unperturbed snaking scenario:

Z(L,ϕ) = Z0(L,ϕ)− λ∗(L) · z(L− ϕ, λ) +O

(

e−2αL)

,

cf. (4.15). By Hypothesis (H4.10), the function z(ϕ, λ) does not vanish at ϕ = ℓm, ℓM . Let us now assume that

z(ϕ, λ) < 0 for all ϕ ∈ S1, λ ∈ Jλ, then the perturbation term in the definition of Z is dominated by λ∗(L) · z(L−ϕ, λ)

for all ϕ ∈ S1, L > L0. Consequently, the greater the term λ∗(L) · z(L− ϕ, λ) the closer lies the zero level set of Z to

4.3 Discussion 53

L

ϕ

Figure 4.5: The left drawing displays the solution of Z = 0 for different perturbation terms in relation to the zerolevel set of Z0. The darker the colour of the isolas the bigger is the underlying perturbation term. At the right thecorresponding isolas in the bifurcation diagram are depicted.

the maxima of the function Z0, cf. Figure 4.5. For “small”L > L0 this perturbation term is comparatively large. But

since λ∗(L)→ 0, when L→∞, the perturbation term decreases and the zero level set of Z approaches the zero level

set of Z0. In the bifurcation diagram this translates into the effect that for increasing L the isolas approach more and

more the primary snakes-and-ladders structure. The same applies for the criss-cross snaking curves. This is sketched

in Figure 4.2 for the case of isolas, as well as, in Figure 4.3 for criss-cross snaking.

µ

λ

µ

λ

Figure 4.6: Snaking within the continuation parameters µ and λ. The left draws the situation for criss-cross snaking,where the (λ, µ) continuation curve is plotted for one (of the two) criss-cross curves. The right panel shows thecorresponding plot in the case of isolas.

Finally we consider the plotting λ versus µ. Also in the (λ, µ) diagram one finds isolas or a sort of snaking, see

Figure 4.6. A similar snaking behaviour can also be found in systems without particular structure, as we consider

them in Chapter 5, cf. Figure 5.5 and Figure 5.11.

55

Chapter 5

Nonreversible snaking

In this chapter we consider a snaking scenario that arises in ODE systems without particular structure, such as first

integrals or reversibility. This distinguishes the present situation from the considerations in Chapter 3 and 4, where

in the unperturbed equation such structures exist. The motivating example for the investigations in this chapter is

system (1.6), whose characteristics are described in the introduction. In particular we give an analytical verification

of the related bifurcation diagram depicted in Figure 1.7, within a more general setup in R3.

Similar to [3] and Chapters 3 and 4 we assume the existence of an organising EtoP cycle, connecting a hyperbolic

equilibrium E with a hyperbolic period orbit P . According to the lack of structure this EtoP cycle is not robust.

Consequently, homoclinic orbits cannot (generically) be continued in one parameter. For that reason we consider a

2-parameter family of differential equations. We denote the family parameter by (µ1, µ2), where (µ1, µ2) are taken

from a closed rectangle J1 × J2. In the spirit of [3] and (Chapters 3 and 4) we demand that the intersection of the

stable and unstable manifolds is described by the graph of a smooth function z = z(ϕ, µ1).

Beside other prerequisites we assume that P has positive Floquet multipliers and show that the bifurcation diagram

of homoclinic orbits looks qualitatively as the bifurcation diagram of (1.6) given in the upper panel of Figure 1.7. In

fact we prove that the continuation curve exists in a subset of J1×J2 and accumulates on a“stripe” in 0, µ2 ⊂ J1×J2.

This chapter is organised as follows. The presentation of our hypotheses and formulation of the snaking result,

Theorem 5.1, is done in Section 5.1. The proof of Theorem 5.1, is carried out in Section 5.2.

After that we consider in Section 5.3 the case, where the periodic orbit P possesses negative Floquet multipliers. We

show that then the bifurcation diagram consists of two continuation curves, accumulating “from both sides” on an

interval in 0, µ2 ⊂ J1 × J2. The results are summarised in Theorem 5.2. Finally we discuss in Section 5.4 the

existence of isolas instead of a snaking scenario. Theorem 5.3 covers these results.

Note that the results of this chapter where already published in [37]. Moreover, Theorem 5.1 was part of the Diploma

thesis [71].

5.1 Setup and main results

We consider a smooth family of differential equations

x = f(x, µ), x ∈ R3, µ = (µ1, µ2) ∈ J1 × J2 ⊂ R2, (5.1)

where J1, J2 are closed intervals, with 0 ∈ intJ1, intJ2.

We assume the following.

56 5 Nonreversible snaking

Hypothesis (H5.1). (i) For all µ ∈ J1 × J2 the vector field f possesses an equilibrium E = x = 0: f(0, µ) ≡ 0;

The equilibrium E is hyperbolic, and its corresponding manifolds satisfy dimWu(E, µ) = 1, dimW s(E, µ) = 2.

(ii) For all µ ∈ J1 × J2 there is a hyperbolic periodic orbit P = γt := γ(t, µ). Further let dimWuuloc (P, µ) = 2,

dimW ssloc(P, µ) = 2. For all µ the minimal period of P is 2π.

(iii) P possesses the Floquet exponents αss(µ) < 0 < αuu(µ).

(iv) There is a maximal interval I2 := [µ2, µ2] ( J2, µ 2

< µ2, such that for µ ∈ 0 × I2 there is a heteroclinic cycle

connecting E and P .

The constant minimal period can always be achieved by an appropriate time transformation. The interval I2 is maximal

in the sense that for (µ1 = 0, µ2) and µ2 > µ2 or µ2 < µ2there is no complete cycle. More precisely with our choice

of dimensions, typically the EtoP connection is of codimension-1 – that means it appears along a curve in parameter

space. This curve is the µ2-axis and the connection splits up when moving off the µ2-axis. On the other hand, the

PtoE connection is typically robust. Nevertheless, by changing parameters within a wider range this connection can

disappear, for instance in the course of a saddle-node bifurcation. These scenarios are made more precise by additional

hypotheses below.

The 3-dimensional state space enforces that both nontrivial Floquet multipliers of P have the same sign.

Hypothesis (H5.2). The nontrivial Floquet multipliers of P are positive.

Observe that in the Chapter 3 and Chapter 4 we also assume the positivity of the Floquet multipliers of P , cf.

Hypotheses (H3.4) and (H4.4). The positivity of the Floquet multipliers determines the Fenichel normal form near

P , cf. Theorem 2.3. However, in Section 5.3, we relax this hypothesis.

The following lemma can be seen as a motivation for our further considerations. Roughly speaking, it says that under

certain transversality conditions on each curve κ intersecting µ1 = 0 transversely, there is a sequence of parameter

values accumulating at µ1 = 0 for which a homoclinic orbit to the equilibrium does exist, see [36, Corollary 4.3].

Lemma 5.1 (Corollary 4.3 in [36]). Assume Hypotheses (H5.1) and (H5.2), and let κ = κ(ι) be a smooth curve in

J1 × J2 intersecting µ1 = 0 transversely in (0, µ2), where µ2 ∈ (µ2, µ2) and κ(0) = (0, µ2). Assume further

(i)⋃

ι

(

Wu(E, κ(ι))× ι)

⋔⋃

ι

(

W ssloc(P, κ(ι))× ι

)

(ii) W s(E, (0, µ2)) ⋔Wuuloc (P, (0, µ2))

Then there is a sequence (ιn), limn→∞

ιn = 0 such that for all µ = κ(ιn), n≫ 1, there is a homoclinic orbit to E.

Assumption (i) of the lemma claims that the extended unstable and stable manifold of the equilibrium and of the

periodic orbit, respectively, intersect transversely, while assumption (ii) claims that the stable and unstable manifold

of the equilibrium and the periodic orbit, respectively, intersect transversely.

Now arises the question whether all κ(ιn) lie on one continuation curve as in our motivating example – see Figure 5.1.

In panel (i) of this figure, the black dots and squares correspond to parameter values on κ for which a homoclinic

orbit exists. The different shapes indicate that the homoclinic orbits are related to different EtoP cycles. Indeed

in [37, Section 6] it is numerically shown that in the intersection of κ with cb there exist two EtoP cycles. This feature

has not been considered in panel (ii).

5.1 Setup and main results 57

µ1

µ2

µ2

µ2

(ii)(i)

κκ

µ1

µ2

hb1

κ(ιn)

cb

Figure 5.1: Homoclinic orbits on a curve κ which intersects the continuation curve of the codimension-1 heteroclinicorbits transversely. Panel (i) is related to the laser model (1.6). The dots and squares indicate that the homoclinicorbits correspond to different EtoP cycles which exist at the intersection point of κ and cb. Panel (ii) visualises thestatement of Lemma 5.1. [37, Figure 5]

Again we transform the vector field into the Fenichel coordinates, see Theorem 2.3:

vc = 1 +Ac(v, µ)vssvuu,

vss = (αss(µ) +Ass(v, µ))vss,

vuu = (αuu(µ) +Auu(v, µ))vuu.

(5.2)

In these coordinates the periodic orbit P is given by the set vss = 0, vuu = 0.

We introduce again sections near P .

Σin := S1 × vss = δ × Iδ, Σout := S1 × Iδ × vuu = δ, (5.3)

cf. Figure 2.1. Further we denote by Wuu(γϕ, µ) the strong unstable fibre of P to the base point γϕ. With that we

define

Γ := (ϕ, µ) ∈ S1 × J1 × J2 |Ws(E, µ) ∩Wuu(γϕ, µ) ∩ Σout 6= ∅. (5.4)

Thus Γ consists of all the tuples (µ, ϕ) for which there exists a PtoE connection that intersects within Σout the strong

unstable fibre Wuu(γϕ, µ) to the base point γϕ = γ(ϕ, µ).

Further, let UΓ be an open neighbourhood of Γ in S1 × J1 × J2.

Hypothesis (H5.3). There is a smooth function g : UΓ → Iδ and an ǫ > 0 such that

(ϕ, vss, δ) ∈W s(E, µ) ∩ Σout | |vss| < ǫ, (ϕ, µ) ∈ UΓ = (ϕ, gss(ϕ, µ), δ) | (ϕ, µ) ∈ UΓ.

As a consequence of that hypothesis, we get that Γ coincides with the zeros of g, see Figure 5.2:

Γ = (ϕ, µ) ∈ S1 × J1 × J2 | gss(ϕ, µ) = 0. (5.5)

Figure 5.2 does already include some specific features of g or Γ, respectively, which we demand in the following

hypothesis:


ϕ

vss

µ2

graph g

UΓ

Γ

Figure 5.2: Visualisation of Hypothesis (H5.3) and its consequence: In this illustration µ1 is fixed with µ1 = 0. Γcoincides with the zeros of g. The graph of g is only drawn for a sample of µ2-values. The dashed lines indicate apossible continuation of g outside of UΓ. [37, Figure 7]

Hypothesis (H5.4). (i) There is a constant b > 0 such that |Dµ2gss(ϕ, µ)| ≥ b, for all (ϕ, µ) ∈ UΓ.

(ii) There is a smooth function z : S1 × J1 → J2 such that Γ = graph z.

We refer to [37, Section 6], where Hypothesis (H5.4), is numerically verified in the laser model equation (1.6). As a

consequence of (5.5) and Hypothesis (H5.4) (ii) we find that in UΓ

gss(ϕ, µ1, µ2) = 0 ⇐⇒ µ2 = z(ϕ, µ1). (5.6)

As a transversality condition for z we assume:

Hypothesis (H5.5). z′(ϕ, 0) = 0 ⇒ z′′(ϕ, 0) 6= 0.

Fix some µ01 close to zero, and let ϕ0 be some value such that z′(ϕ0, µ0

1) = 0. Using this, we define µ02 := z(ϕ0, µ0

1),

and µ0 := (µ01, µ

02). Furthermore, considering the derivatives of gss(·, µ0

1, z(·, µ01)) at ϕ = ϕ0 we find with (5.6) and

Hypothesis (H5.5) that

Dϕgss(ϕ0, µ0) = 0, D2

ϕgss(ϕ0, µ0) 6= 0. (5.7)

Note that graph gss(·, µ0) describes the stable manifold W s(E, µ0) near ϕ0. Therefore (5.7) means that W s(E, µ0)

and Wuuloc (P, µ

0) have a quadratic tangency in ϕ0. We refer to Figure 5.3 for an illustration of the consequence of

Hypothesis (H5.5).

Let the function z(·, µ1) take its minimum in ℓm(µ1), and similarly let z(·, µ1) be maximal in ℓM (µ1). This defines

functions µ1,min : µ1 7→ z(ℓm(µ1), µ1) and µ1,max : µ1 7→ z(ℓM (µ1), µ1), both mapping J1 → J2. The graphs of these

functions define the µ-region for which a heteroclinic cycle connecting E and P exists. In our motivating example this

region is just the stripe between the two curves tb – see Figure 1.7. Hence, the maximal interval [µ2, µ2] defined in

Hypothesis (H5.1) is given by

µ2:= z(ℓm(0), 0), µ2 := z(ℓM (0), 0).

Moreover, for each µ between the graphs of µ1,min and µ1,max, there are at least two heteroclinic PtoE connections.

These undergo saddle-node bifurcations on the graphs of µ1,min and µ1,max. In particular, moving along the µ2-axis

the heteroclinic PtoE connections undergo saddle-node bifurcations in µ2and µ2. If z has exactly one minimum (and

5.1 Setup and main results 59

vss

µ2

Σout

graph g =W s(E, µ)

ϕ

Γ = graph z

Wuuloc (P, µ)

ϕ0

µ2

Figure 5.3: Quadratic tangency of Wuuloc (P, µ) and W s(E) as a consequence of Hypothesis (H5.5): In this illustration

µ1 is fixed with µ1 = µ1. For µ1 = 0, this is an enlargement of a detail around the turning point of Γ in Figure 5.2. [37,Figure 8]

hence one maximum), there are exactly two heteroclinic PtoE connections between the graphs of µ1,min and µ1,max

– see Figure 5.2.

Next we consider the EtoP connection.

Hypothesis (H5.6). There exist smooth functions huu : J1 × J2 → Iδ, hc : J1 × J2 → S1 such that

(vc, δ, vuu) ∈Wu(E, µ) ∩ Σin |µ ∈ J1 × J2 = (vc, δ, vuu) = (hc(µ), δ, huu(µ)) |µ ∈ J1 × J2.

Moreover,

(i) huu(0, µ2) ≡ 0, and ∀µ2 ∈ J2 holds Dµ1huu(0, µ2) 6= 0,

(ii) ∃q < 1 : ∀ϕ ∈ S1 | ddϕhc(0, z(ϕ, 0))| ≤ q.

Σin

(hc(µ1, µ2), huu(µ1, µ2)) : µ1 ∈ U(0)

(hc(0, µ2), huu(0, µ2))

vuu

vc

Wuuloc (P )

Figure 5.4: A visualisation of Hypothesis (H5.6) (i) with µ2 = µ2: The curve (hc(µ1, µ2), huu(µ1, µ2)) =⋃

µ1

Wu(E, µ1, µ2) intersects Wssloc(P ) transversely. [37, Figure 9]

By definition

Wu(E, µ) ∩W ssloc(P, µ) ∩ Σin 6= ∅ ⇐⇒ huu(µ) = 0. (5.8)

So, Hypothesis (H5.6) (i) says that for all µ on the µ2-axis, there is a heteroclinic orbit connecting E to P . In other

words, the µ2-axis is on a par with the curve cb of our motivating example – see Figure 1.7 or Figure 5.1, respectively.

Moreover, moving through the µ2-axis transversely effects that the EtoP connection splits up with nonzero speed

– see also Lemma 5.1. The consequences of Hypothesis (H5.6) (i) for the shape and mutual position of the traces


of Wu(E, µ) and W ssloc(P, µ) are depicted in Figure 5.4. Finally, we note that by this assumption Dµ1h

uu(0, µ2) is

bounded away from zero.

Recall that z(ϕ, 0) determines the µ2 values for which a EtoP cycle exists (clearly µ1 = 0), where ϕ is the vc-coordinate

value of the intersection of the corresponding PtoE connection with Σout. Whereas hc(0, z(ϕ, 0)) is the vc-coordinate

value of the intersection of the corresponding EtoP connection with Σin. Hence, Hypothesis (H5.6) (ii) yields that the

proportion of the alteration rates of these vc-coordinates is bounded by q < 1. In other words, these coordinate values

must not move against each other too fast. It will be used in the next section for solving the bifurcation equations.

Now we can state our main result guaranteeing a snaking scenario.

Theorem 5.1. † Assume Hypotheses (H5.1)–(H5.6). Then there is a constant L0 > 0, and there are functions

µi : (L0,∞)→ R, i = 1, 2, such that for each L > L0 there is a homoclinic orbit to E for µ ∈ J1× J2 that spends time

2L between Σin and Σout if and only if µ = (µ1(L), µ2(L)).

Moreover there are an α > 0, a 2π-periodic function ϕ∗0(·) and a positive bounded function au such that

µ1(L) =au(L)

Dµ1huu(0,z(ϕ∗

0(L)+2L,0))e−2αuu(0,z(ϕ∗

0(L)+2L,0))L(1 +O(e−αL)),

µ2(L) = z(ϕ∗0(L) + 2L, 0) +O(e−αL).

It follows immediately that µ1(L) tends to zero as L goes to infinity. Further it is obvious that µ2(L) is a perturbation

of z(ϕ∗0(L) + 2L, 0). This result resembles pretty much the statement about the snaking parameter µ in original

snaking scenario, cf. Theorem 2.1. But here, in contrast to Theorem 2.1, the term ϕ∗0(L) is periodic and not constant.

If ϕ∗0(L) + 2L is monotonically increasing, then µ2(·) essentially copies the behaviour of z(·, 0), see Figure 5.5 and

Lemma 5.2.

Λ(L)

µ1

z(ϕ, 0)

µ2

ϕ

I2

µ2

Figure 5.5: The relation between graph z and the snaking curve Λ(L) = (µ1(L), µ2(L)). The shape of graph z(·, 0)depicted in the left panel is passed on to the snaking curve Λ(L) in the right panel. The snaking curve accumulatesat 0× I2, the set of parameters for which the primary EtoP cycle exits, see Hypothesis (H5.1) (iii). [37, Figure 10]

In the following lemma, we describe the shape of the snaking curve (µ1(L), µ2(L)) somewhat closer. We consider z(·, 0)

as a periodic function R→ R. We denote the first and second derivative of µ2 by µ′2 and µ′′

2 , respectively.

† Note that Theorem 5.1 was already published in the Diploma thesis [71].

5.2 Snaking analysis 61

Lemma 5.2. Assume Hypotheses (H5.1)–(H5.6) with the more severe condition q < 1/2, see Hypothesis (H5.6) (ii).

Then Φ : L 7→ ϕ∗0(L) + 2L is a transformation, and for each ϕ with z′(ϕ, 0) = 0 exists a unique L in a small

neighbourhood of Φ−1(ϕ) such that µ′2(L) = 0. Moreover µ′′

2(L) 6= 0. These are the only zeros of µ′2.

The proofs of these statements are carried out in Section 5.2. Prior to that, however, we give a geometric explanation

with the help of the Figure 5.6. Assume that the unstable manifold of the equilibrium depends only on µ1, and similarly

that the stable manifold of the equilibrium depends only on µ2: Wu(E, µ) = Wu(E, µ1), W s(E, µ) = W s(E, µ2).

In Figure 5.6 (i), we consider a fixed Poincare section of P . Fix some µ2 – and therefore one particular position of

W s(E, µ2) – and assume that an increasing µ2 effects upward motion of W s(E, µ2). The bullet defines a µ1 for which

W s(E, µ2) and Wu(E, µ1) intersect and therefore a homoclinic orbit to E does exist. This homoclinic orbit can be

continued by moving W s(E, µ2) up and down. The corresponding continuation curve of the bullet in the µ-space is

displayed Figure 5.6 (ii).

graphµ1,max

graphµ1,min

µ1

µ2

(i) (ii)

via µ2

⋃

µ1∈J1

Wu(E, µ1)

W s(E, µ2)P

Figure 5.6: The creation of a snaking curve: Panel (i) shows part of a (global) Poincare section containing both Eand P . Panel (ii) shows the continuation curve of homoclinic orbits to E. [37, Figure 11]

Discussion of the hypothesis

Before proving Theorem 5.1 we justify that the assumptions stated in this section are ”sensible”. In [37, Section

6] the motivating example (1.6) has been numerical investigated by Rieß and it was shown that this system meets

Hypothesis (H5.1), Hypothesis (H5.6). To justify Hypotheses (H5.3) and (H5.5) Rieß calculated the shape of the

function Γ, that is depicted in Figure 5.7 below. Comparing this computation with the assumption on Γ that are

sketched in Figure 5.2, it turns out that example (1.6) satisfies these hypotheses.

5.2 Snaking analysis

This section is devoted the proof of Theorem 5.1. As in Section 3.2.1 we find that homoclinic solutions of (5.1)

correspond to solutions v(·, µ) of (5.2) with

v(−L, µ) ∈Wu(E, µ) ∩ Σin, v(L, µ) ∈W s(E, µ) ∩ Σout. (5.9)

As before these conditions lead to a Shilnikov Problem. But in contrast to Chapter 3 and Chapter 4, we do not use

Theorem 2.4 to solve the Shilnikov problem, but the following corollary of Lemma 2.2, which takes into account that

the periodic orbit may have negative Floquet multipliers.


-1.466

-1.464

-1.462

-1.46

-1.458

0 π/2 π 3π/2 2πϕ

ν2

Γ

Figure 5.7: The computed curve Γ for system (1.6). The curve is shown as an ϕ-vs-µ2 = ν2 plot, where the angle ϕcorresponds to used in Hypothesis (H5.4). [37, Figure 16]

Corollary 5.3 (Shilnikov for negative Floquet multipliers). There is a positive constant L0 such that for all L > L0,

all (ϕ, µ) ∈ S1 × J1 × J2, and χs, χu ∈ ±1 there exists a unique solution v(t), also referred to as v(t, ϕ, µ, χs, χu),

of (5.2) with

vss(−L) = χsδ, vc(−L) = ϕ and vuu(L) = χuδ.

Moreover there is a positive constant α < minµ∈J1×J2|αss(µ)|, αuu(µ) such that

v(−L) =(

ϕ, χsδ, χu∆uue−2αuu(µ)L (1 +O(e−αL))

)

,

v(L) =(

ϕ+ 2L+O(e−αL), χs∆sse2α

ss(µ)L (1 +O(e−αL)), χuδ)

,(5.10)

where ∆ss and ∆uu are positive functions depending on (ϕ, µ, χs, χu). Moreover, ∆ss/uu(·, ·, χs, χu) are smooth. The

derivatives of v satisfy the following estimates

Dz1...zjv(−L) = (Dz1...zjϕ, 0, χuDz1...zj (∆uue−2αuu(µ)L) (1 +O(e−αL))),

Dz1...zjv(L) = (Dz1...zj (ϕ+ 2L) +O(e−αL), χsDz1...zj (∆sse2α

ss(µ)L) (1 +O(e−αL)), 0).(5.11)

Here zi ∈ L, µ, ϕ for i = 1, . . . , j and j ∈ 1, 2, 3.

Since for fixed χs, χu, the functions ∆ss and ∆uu are defined on the compact set S1×J1×J2, they are bounded away

from zero. Further we observe that ∆ss and ∆uu are independent of L.

In the present situation P possesses positive Floquet multipliers. Therefore the mere difference between Lemma 2.2

and Corollary 5.3 is that the vc-component is determined to be ϕ at −L instead of 0. This is due to the fact that in

contrast to [3] or Chapters 3 and 4 the considered ODE possesses no reversible structure for all values of µ. Hence

there is no advantage to define vc in the middle of the interval [−L,L]. In fact defining vc at −L makes the following


analysis a bit more “convenient”. The more general setting of Corollary 5.3 is used later in Section 5.3.

The proof of Corollary 5.3 proceeds basically as the proof of Theorem 2.4.

Proof of Theorem 5.1. To describe the transition from Σin to Σout we use the function v defined by Corollary 5.3.

By our choice of Σin and Σout, see (5.3), this transition is determined by v(·, ϕ, µ) := v(·, ϕ, µ, 1, 1). Using the notation

introduced in Section 5.1, Equation (5.9) translates to

vc(−L,ϕ, µ) = hc(µ), (5.12)

vuu(−L,ϕ, µ) = huu(µ), (5.13)

vss(L,ϕ, µ) = gss(vc(L,ϕ, µ), µ). (5.14)

As in Chapter 3 and Chapter 4 we are interested in those homoclinic orbits which are in a small neighbourhood of a

heteroclinic cycle. These cycles are determined by huu = 0 and gss = 0, see (5.8) and (5.5). For that reason we solve

(5.12)–(5.14) near huu = 0 and gss = 0.

In accordance with (5.10) we find that vc(L,ϕ, µ) = ϕ+ 2L+O(e−αL). Motivated by this equality, we introduce the

following time transformation

l = L+O(e−αL). (5.15)

Indeed, (5.15) can be solved for

L = L∗(l, µ) = l +O(e−αl). (5.16)

Using this, Equation (5.14) can be rewritten as vss(l+O(e−αl), µ) = gss(ϕ+ 2l, µ). Altogether, using the new time l

and the estimates (5.10) the system (5.12)–(5.14) reads

ϕ = hc(µ1, µ2), (5.17)

∆uue−2αuu(µ1,µ2)l(1 +O(e−αl)) = huu(µ1, µ2), (5.18)

∆sse2αss(µ1,µ2)l(1 +O(e−αl)) = gss(ϕ+ 2l, µ1, µ2). (5.19)

First, we consider (5.19), which describes the coupling in Σout. Recall that we want to solve (5.14), and therefore also

(5.19), near g = 0. Furthermore, recall from (5.6) that gss(ϕ+ 2l, µ1, z(ϕ+ 2l, µ1)) ≡ 0. Now write

µ2 = z(ϕ+ 2l, µ1) + e

and expand gss(ϕ+ 2l, µ1, z(ϕ+ 2l, µ1) + e) with respect to e. Inserting in (5.19) gives

∆sse2αss(µ1,z(ϕ+2l,µ1)+e)l(1 +O(e−αl)) = Dµ2g

ss(ϕ+ 2l, µ1, z(ϕ+ 2l, µ1)) · e+O(e2).

Using this and Hypothesis (H5.4), the coupling equation (5.14) eventually reads:

∆ss(ϕ,µ1,z(ϕ+2l,µ1)+e)Dµ2g

ss(ϕ+2l,µ1,z(ϕ+2l,µ1))e2α

ss(µ1,z(ϕ+2l,µ1)+e)l(1 +O(e−αl)) = e+O(e2). (5.20)

For |e| ≪ 1, sufficiently large l and all ϕ this equation can be solved for e = e∗(l, ϕ, µ1) by means of the Implicit


Function Theorem. The solving function e∗ is differentiable. Further we see from (5.20) that

e∗(l, ϕ, µ1) =∆ss(ϕ,µ1,z(ϕ+2l,µ1))

Dµ2gss(ϕ+2l,µ1,z(ϕ+2l,µ1))

e2αss(µ1,z(ϕ+2l,µ1))l(1 +O(e−αl)). (5.21)

Altogether we find that the coupling equation (5.14) can be solved for µ2 = µ2(l, ϕ, µ1) with

µ2(l, ϕ, µ1) = z(ϕ+ 2l, µ1)) + e∗(l, ϕ, µ1),

where the leading order term of e∗ is given by (5.21).

Now we turn towards the coupling in Σin which is determined by (5.17) and (5.18). Using the representation of µ2,

these equations read

ϕ = hc(µ1, z(ϕ+ 2l, µ1) + e∗(l, ϕ, µ1)), (5.22)

∆uue−2αuu(µ1,z(ϕ+2l,µ1)+e∗(l,ϕ,µ1))l(1 +O(e−αl)) = huu(µ1, z(ϕ+ 2l, µ1) + e∗(l, ϕ, µ1)). (5.23)

We solve (5.22), (5.23) for (ϕ, µ1) depending on l. Note that ϕ is the vc coordinate where the prospective homoclinic

orbits hits Σin. Hence, ϕ may vary within a “large” range. To handle this difficulty analytically, we first consider

the “unperturbed equation” ϕ = hc(0, z(ϕ + 2l, 0)). For that we consider z(·, 0) as a 2π-periodic function R → R,

see Hypothesis (H5.1) and Hypothesis (H5.4). Hence, hc(0, z(· + 2l, 0)) is a 2π-periodic function as well. Because of

Hypothesis (H5.6) (ii), we can apply again the Implicit Function Theorem to find a unique solution ϕ∗0(l) on R such

that

ϕ = hc(0, z(ϕ+ 2l, 0)) ⇐⇒ ϕ = ϕ∗0(l). (5.24)

Note that ϕ∗0(·) is again 2π-periodic. Now, write

ϕ = ϕ∗0(l) + ψ,

and we define Hc(l, ψ, µ1), Huu(l, ψ, µ1) by

Hc/u(l, ψ, µ1) := hc/u(µ1, z(ϕ∗0(l) + ψ + 2l, µ1)).

Using these terms the right-hand sides of (5.22) and (5.23) read

hc/u(µ1, z(ϕ∗0(l) + ψ + 2l, µ1) + e∗(l, ϕ∗

0(l) + ψ, µ1)) = Hc/u(l, ψ, µ1) + rc/u(e∗(l, ϕ∗0(l) + ψ, µ1)),

where rc/u(e∗) = O(e∗). Further, since ϕ∗0 is the unique solution of (5.24), we find

Hc(l, 0, 0) = hc(0, z(ϕ∗0(l) + 2l, 0)) = ϕ∗

0(l).

In accordance with Hypothesis (H5.6) (i), we find furthermore

Huu(l, 0, 0) = 0, DψHuu(l, 0, 0) = 0.


Hence, (5.22) and (5.23) are equivalent to

ψ = DψHc(l, 0, 0)ψ +Dµ1

Hc(l, 0, 0)µ1 +O(|(ψ, µ1)|2) + rc(e∗), (5.25)

∆uue−2αuu(l,ψ,µ1)l(1 +O(e−αl)) = Dµ1Huu(l, 0, 0)µ1 +O(|(ψ, µ1)|

2) + ru(e∗), (5.26)

with αuu(l, ψ, µ1) := αuu(µ1, z(ϕ∗0(l) + ψ + 2l, µ1) + e∗(l, ϕ∗

0(l) + ψ, µ1)).

Our goal is now to solve the system (5.25), (5.26) for (ψ, µ1) depending on l. To this end we invoke again the Implicit

Function Theorem. The main observation is that due to Hypothesis (H5.6)

|DψHc(l, 0, 0)| ≤ q < 1, Dµ1H

uu(l, 0, 0) 6= 0.

Note that due to (5.11), the corresponding partial derivatives of rc(e∗(l, ϕ∗0(l)+ψ, µ1)) and r

u(e∗(l, ϕ∗0(l)+ψ, µ1)) tend

to zero as l→∞. Therefore there exist unique functions ψ∗(l), µ∗1(l) satisfying the system (5.25), (5.26). Accordingly,

(5.22), (5.23) are satisfied by

φ∗(l) := ϕ∗0(l) + ψ∗(l) and µ∗

1(l).

Inspecting (5.23) and (5.25), we find with au(l) := ∆uu(ϕ∗0(l), 0, z(ϕ

∗0(l), 0)) that

µ∗1(l) =

au(l)Dµ1H

uu(l,0,0)e−2αuu(l,0,0)l(1 +O(e−αl)) and ψ∗(l) = O(e−2αuu(l,0,0)l).

Altogether, for (5.17)–(5.19) we find the unique solution (ϕ, µ1, µ2)(l) = (φ∗(l), µ∗1(l), µ

∗2(l)), where

µ∗2(l) := µ2(l, φ

∗(l), µ∗1(l)) = z(φ∗(l) + 2l, µ∗

1(l)) + e∗(l, φ∗(l), µ∗1(l))

= z(ϕ∗0(l) + 2l, 0) +O(e−αl).

Note that v spends the time 2l+O(e−αl) between Σin and Σout, see (5.15). So, in view of the statement in Theorem 5.1,

we define

φ(L) := φ∗(l(L)), µ1(L) := µ∗1(l(L)), µ2(L) := µ∗

2(l(L)).

Then (ϕ, µ1, µ2)(L) = (φ(L), µ1(L), µ2(L)) solves (5.12)–(5.14). The above considerations yield

φ(L) = ϕ∗0(L) +O(e

−αL),

µ1(L) =au(L)

Dµ1Huu(L,0,0)e

−2αuu(L,0,0)L(1 +O(e−αL)),

µ2(L) = z(ϕ∗0(L) + 2L, 0) +O(e−αL).

This finally completes the proof of Theorem 5.1.

Proof of Lemma 5.2. In what follows, we sketch the proof of Lemma 5.2. We note that, due to (5.11), the O-term

in the representation of µ2(L) is differentiable and its derivative can be estimated by a O-term of the same order. The

same holds true for higher derivatives. Therefore we find

µ′2(L) = z′(ϕ∗

0(L) + 2L, 0)(ϕ∗0′(L) + 2) +O(e−αL).


Recall the determining equation ϕ∗0(L) = hc(0, z(ϕ∗

0(L) + 2l, 0)) for ϕ∗0(L), see (5.24). From this equation, we get an

estimate of the derivative of ϕ∗0(L), whereby we finally confirm that for sufficiently large L

ϕ∗0′(L) + 2 6= 0.

So, necessarily the zeros of µ′2(L) are close to the zeros of z′(ϕ∗

0(L) + 2L, 0)). Let z′(ϕ∗0(L0) + 2L0, 0)) = 0. Using the

contraction principle we find a neighbourhood U(L0) of L0 in which µ′2(L) = 0 has a unique solution L. Straightforward

computations show µ′′2(L) 6= 0. Further, the size of the neighbourhood U(L0) can be chosen independently of L0.

Outside the union of these neighbourhoods, z′(·, 0) is bounded away from zero. This finally shows that outside the

union of these neighbourhoods µ′2(L) has no zeros for sufficiently large L.

5.3 Negative Floquet multipliers

In this section we discuss the scenario, where the nontrivial Floquet multipliers of the periodic orbit P are negative –

in other words, we replace Hypothesis (H5.1) by:

Hypothesis (H5.7). The nontrivial Floquet multipliers of P are negative.

Since our setting is in R3, the two nontrivial Floquet multipliers of P must have the same sign. Negative multipliers

cause that the vector bundle consisting of the eigenvectors of the monodromy matrices along P is a Mobius strip

and thus not orientable. Hence, we cannot introduce Fenichel coordinates near P . We overcome this difficulty by

introducing local coordinates, which are not 2π periodic, but of period 4π. To this end we transform at first (5.1) into

normal form, see [65, Theorem 3.11] or the corresponding transformations in Section 6.2, which gives:

θ = 1,

y = B(θ)y + F (θ, y, µ),

where y = (y1, y2) ∈ R2 and θ ∈ S1; B and F are smooth. Furthermore we straighten the stable and unstable fibres

of P , as it is done in Section 6.2. This yields that the function F satisfies

F (θ, 0, y2, µ) ≡ 0 and F (θ, y1, 0, µ) ≡ 0 and DyF (θ, 0, 0, µ) ≡ 0, ∀µ ∈ J1 × J2, ∀θ ∈ S1.

After that we apply Floquet theory to the linear system y = B(θ)y, see [65, Theorem 3.12] for more details. This

transforms the above normal form into:

vc = 1,

˙vss = αss(µ)vss + F ss(vc, vss, vuu, µ), (5.27)

˙vuu = αuu(µ)vuu + Fuu(vc, vss, vuu, µ),

where vc ∈ S1 := R/∼4π, and x ∼4π y ⇔ x = y mod4π. Moreover, similar arguments as leading to Theorem 2.3 yield

F ss(vc, 0, vuu, µ) = Fuu(vc, vss, 0, µ) = 0, as well as, Dvss Fss(vc, 0, 0, µ) = Dvuu Fuu(vc, 0, 0, µ) = 0.

Note that by this construction the two points (vc, vss, vuu) and (vc + 2π,−vss,−vuu) represent the same point in

5.3 Negative Floquet multipliers 67

(θ, y)-coordinates. In other words, two points are identified via the map:

i : S1 × Iδ × Iδ → S1 × Iδ × Iδ

(vc, vss, vuu) 7→ (vc + 2π,−vss,−vuu).

Next, in accordance with the procedure in Section 5.1, we introduce a cross-section Σin of W ssloc(P, µ) intersecting

orthogonally the stable fibres of P in a distance δ of P . Similarly we define Σout. In (vc, vss, vuu)-coordinates these

sections read:

Σin+ := S1 × vss = δ × Iδ, Σout+ := S1 × Iδ × vuu = δ. (5.28)

The subscript “+” refers to the positive value δ for the fixed vss- and vuu-coordinate, respectively. The sections defined

in (5.28) are identified via the map i with

Σin− := S1 × vss = −δ × Iδ, Σout− := S1 × Iδ × vuu = −δ.

Further we introduce a set Γ similarly to (5.4) – formally replacing S1 by S1:

Γ := (ϕ, µ) ∈ S1 × J1 × J2 |Ws(E, µ) ∩Wuu(γϕ, µ) ∩ Σout+ 6= ∅. (5.29)

Note that each strong unstable fibre Wuu(γϕ, µ) of P intersects Σout twice. In the terminology of (5.29) those two

points are represented by Wuu(γϕ, µ) ∩ Σout+ and Wuu(γϕ+2π, µ) ∩ Σout+ .

Let UΓ be an open neighbourhood of Γ in S1 × J1 × J2. Regarding the PtoE connecting orbit we assume:

Hypothesis (H5.8). There is a smooth function gss : UΓ → Iδ and an ǫ > 0 such that

(ϕ, vss, δ) ∈W s(E, µ) ∩ Σout+ | |vss| < ǫ, (ϕ, µ) ∈ UΓ = (ϕ, gss(ϕ, µ), δ) | (ϕ, µ) ∈ UΓ.

Hypothesis (H5.9). (i) There is a constant b > 0 such that |Dµ2gss(ϕ, µ)| ≥ b, for all (ϕ, µ) ∈ UΓ.

(ii) There is a smooth function z : S1 × J1 → J2 such that Γ = graph z.

Consequently

gss(ϕ, µ1, z(ϕ, µ1)) ≡ 0.

Hypothesis (H5.10). z′(ϕ, 0) = 0 ⇒ z′′(ϕ, 0) 6= 0.

Finally, regarding the EtoP connecting orbit we assume:

Hypothesis (H5.11). There exist smooth functions huu : J1 × J2 → Iδ, hc : J1 × J2 → S1 such that

(vc, δ, vuu) ∈Wu(E, µ) ∩ Σin+ , µ ∈ J1 × J2 = (vc, δ, vuu) = (hc(µ), δ, huu(µ)), µ ∈ J1 × J2.

Moreover,

(i) huu(0, µ2) ≡ 0, and ∀µ2 ∈ J2 holds Dµ1huu(0, µ2) > 0,

(ii) ∃q < 1 : ∀ϕ ∈ S1 | ddϕhc(0, z(ϕ, 0))| ≤ q.


Indeed, in Hypothesis (H5.11)(i) it already suffices to assume Dµ1huu(0, µ2) 6= 0. The specification stated in the

hypothesis determines the sign of the functions µ+1 and µ−

1 in the way as stated in theorem below.

Now, the analogue of Theorem 5.1 reads:

Theorem 5.2. Assume Hypothesis (H5.1), and Hypotheses (H5.7)–(H5.11). Then there is a constant L0 > 0, and

there are functions µ+i , µ

−i : (L0,∞) → R, i = 1, 2, such that for each L > L0 there is a homoclinic orbit to E for

µ ∈ J1 × J2 that spends time 2L between Σin and Σout if and only if µ = (µ±1 (L), µ

±2 (L)).

Moreover there are an α > 0, two 4π-periodic functions ϕ+0 (·), ϕ

−0 (·) and positive bounded functions a+u , a

−u such that

µ±1 (L) =

±a±u (L)

Dµ1huu(0,z(ϕ±

0 (L)+2L,0))e−2αuu(0,z(ϕ±

0 (L)+2L,0))L(1 +O(e−αL)),

µ±2 (L) = z(ϕ±

0 (L) + 2L, 0) +O(e−αL).

A visualisation of the statement of this theorem is given in Figure 5.8.

I2

µ2

Λ+(L)Λ−(L)

µ1

z(ϕ, 0)

µ2

ϕ

Figure 5.8: As in the case of positive multipliers the shape of graph z(·, 0) is passed on to the snaking curves Λ±(L),see Figure 5.5. The snaking curves accumulate at 0 × I2 from different sides. [37, Figure 12]

Proof of Theorem 5.2. We pursue the same strategy as in the proof of Theorem 5.1: We construct homoclinic

orbits to E by coupling in Σin and Σout the unstable and stable manifolds, respectively, with solutions according to

Corollary 5.3. However, here there are two possibilities for the transition from Σin to Σout. There are solutions of

(5.27) starting in Σin+ ∩ vuu > 0 and end up in Σout+ ∩ vss > 0, and there are solutions starting in Σin− ∩ v

uu > 0

and end up in Σout+ ∩ vss < 0, see. Figure 5.9. In the language of Corollary 5.3 this distinction is determined by the

signs of χs and χu.

The transition Σin+ ∩ vuu > 0 to Σout+ ∩ vss > 0

Here we employ solutions of Corollary 5.3 with (χs, χu) = (1, 1) what we suppress from the notation. In this case

the argumentation runs completely parallel to the proof of Theorem 5.1. We confine to sketch the procedure: The

coupling equations analogue to (5.12)–(5.14) are almost the same:

vc(−L,ϕ, µ) = hc(µ),

vuu(−L,ϕ, µ) = huu(µ),

vss(L,ϕ, µ) = gss(vc(L,ϕ, µ), µ),

5.3 Negative Floquet multipliers 69

ϕ4π

2π

P

W ssloc(P )

Wuuloc (P )

Σin+

Σout+

Σin−

vuuδ

vss

δ

−δ

Figure 5.9: The cross-sections Σin and Σout. In Σin+ and Σin

− there is drawn a curve (hc, huu)(·, µ2) for fixed µ2, seeHypothesis (H5.11). In Σout

+ there is drawn the graph of gss(·, µ) for fixed µ, see Hypothesis(H5.8). [37, Figure 13]

with the only difference that here ϕ, vc ∈ S1. From that we gain the analogue to (5.17)–(5.19)

ϕ = hc(µ),

∆uue−2αuu(µ)l(1 +O(e−αl)) = huu(µ), (5.30)

∆sse2αss(µ)l(1 +O(e−αl)) = gss(ϕ+ 2l, µ).

Proceeding in the same way as in the proof of Theorem 5.1 we get the solutions µ+1 (L) and µ+

2 (L) as stated in the

theorem. The function ϕ+0 solves the analogue to (5.24).

The transition Σin− ∩ vuu > 0 to Σout+ ∩ vss < 0

First we observe that the intersections of Wu(E, µ) with Σin written in the form (hc(µ), δ, huu(µ)) ∈ Σin+ , see Hypoth-

esis (H5.11), are identified with (hc(µ) + 2π,−δ,−huu(µ)) ∈ Σin− . This allows to employ solutions of Corollary 5.3

with (χs, χu) = (−1, 1) for our analysis. Again we suppress the χs, χu-dependence from the notation.

Thus the coupling equations read:

vc(−L,ϕ, µ) = hc(µ) + 2π,

vuu(−L,ϕ, µ) = −huu(µ),

vss(L,ϕ, µ) = gss(vc(L,ϕ, µ), µ).

With the results of Corollary 5.3 this can be rewritten as

ϕ = hc(µ) + 2π,

∆uue−2αuu(µ)l(1 +O(e−αl)) = −huu(µ), (5.31)

−∆sse2αss(µ)l(1 +O(e−αl)) = gss(ϕ+ 2l, µ).


Now we can proceed again as in the proof of Theorem 5.1, with the minor difference that the function ϕ−0 results from

the fixed point equation

ϕ = hc(0, z(ϕ+ 2l, 0)) + 2π.

This finally leads to the solutions µ−1 (L) and µ

−2 (L).

We want to note that µ+1 and µ−

1 have different signs. This follows immediately from (5.30) and (5.31) in combination

with Hypothesis (H5.11)(i).

5.4 Isolas

In this Section we consider a further continuation scenario of homoclinic orbits. We retain the general assumptions

of Hypothesis (H5.1), and suppose positive Floquet multipliers. Further we adopt the definition of Γ, see (5.4), and

assume Hypothesis (H5.3). In Section 5.2, we have seen that the shape of the continuation curve (µ1(L), µ2(L)) is

basically determined by the form of the set Γ, which was assumed to be the graph of a function z : S1 × J1 → J2.

Now we investigate the consequences of altering the corresponding Hypothesis (H5.4). More precisely, we suppose

that Γ is no longer graph of a function, but for fixed µ1 a closed curve in each case.

In contrast to the situation in Section 5.2, we end up with a sequence of closed continuation curves, i.e. isolas.

Consequently, the homoclinic orbits that are close to the primary EtoP cycle do not lie on one (global) continuation

curve. The addressed curves tend to 0 × I2 in the sense of the Hausdorff metric.

A similar scenario was already discussed in [3] for the Hamiltonian case under slightly different assumptions.

We assume the following:

Hypothesis (H5.12). Let I1 ( J1 be a closed interval containing zero and let Iϕ ( S1. There exist smooth functions

ϕ : S1 × I1 → Iϕ and µ2 : S1 × I1 → J2 such that

Γ = (ϕ(r, µ1), µ1, µ2(r, µ1)) | r ∈ S1, µ1 ∈ I1,

where

(Drϕ(r, µ1), Drµ2(r, µ1)) 6= 0, ∀r ∈ S1, ∀µ1 ∈ I1.

Hence, the set Γ ∩ µ1 = const. is a closed, regular curve in Iϕ × J2 parametrised by some parameter r ∈ S1.

ϕ

vssµ2

UΓ

Γ

Figure 5.10: Visualisation of Hypothesis (H5.12). Depicted is the closed curve (ϕ(r, 0), µ1, µ2(r, 0)) | r ∈ S1 =gss(·, 0, ·) = 0. [37, Figure 14]

Again we denote by UΓ an open neighbourhood of Γ in S1× J1× J2. Further, (5.5) together with Hypothesis (H5.12)

5.4 Isolas 71

yield as counterpart to (5.6)

gss(ϕ(r, µ1), µ1, µ2(r, µ1)) ≡ 0, ∀ (r, µ1) ∈ S1 × I1. (5.32)

Hypothesis (H5.13).

Dϕgss(ϕ, 0, µ2)Drµ2(r, 0)−Dµ2g

ss(ϕ, 0, µ2)Drϕ2(r, 0) 6= 0.

Assume Hypothesis (H5.12). Then Hypothesis (H5.13) means that the gradient of gss(·, 0, ·) does not vanish at any

point within the set Γ.

Hypothesis (H5.14).

ϕ(r, 0)− hc(0, µ2(r, 0)), r ∈ S1 ( S1.

The subtraction in the hypothesis is done in S1. Since Iϕ is a proper subset of S1, this hypothesis is satisfied if

|Dµ2hc(0, µ2(r, 0))Drµ2(r, 0)| ≪ 1, ∀ r ∈ S1.

Theorem 5.3. Assume Hypotheses (H5.1)–(H5.3), (H5.6) (i) and Hypotheses (H5.12)–(H5.14). Then, there is a

sequence of mutually disjoint closed continuation curves Λk := (µ1,k(r), µ2,k(r)), r ∈ S1, k ∈ N. These curves tend

towards a segment of the µ2-axis in the sense of the Hausdorff metric. More precisely, for sufficiently large k ∈ N,

there exist mutually disjoint intervals Ik ⊂ R and smooth functions µi,k : S1 → Ji, i = 1, 2, and Lk : S1 → Ik such

that there is a homoclinic orbit to E for µ ∈ J1 × J2 with flight time L ∈ Ik from Σin to Σout, if and only if, there

exists an r ∈ S1 such that µ = (µ1,k(r), µ2,k(r)) and L = Lk(r). Moreover

µ1,k(r) = O(e−αk)

µ2,k(r) = µ2(r, µ1,k(r)) +O(e−αk).

µ1

µ2

I2

Λk+1Λk

Figure 5.11: Visualisation of Theorem 5.3: The continuation curves Λk. [37, Figure 15]

Proof of Theorem 5.3.

We follow the lines of the proof of Theorem 5.1 up to Equations (5.17) – (5.19), which we repeat here:

ϕ = hc(µ1, µ2), (5.33)

∆uue−2αuu(µ1,µ2)l(1 +O(e−αl)) = huu(µ1, µ2), (5.34)

∆sse2αss(µ1,µ2)l(1 +O(e−αl)) = gss(ϕ+ 2l, µ1, µ2). (5.35)


Again we look for solutions of these equations near the set Γ. For sufficiently small µ1 ∈ J1 each point of a small

tubular neighbourhood of Γ ∩ µ1 = µ1 has a unique representation:

(ϕ+ 2l) mod 2π = ϕ(r, µ1) + ϕ

µ2 = µ2(r, µ1) + µ2,

where (ϕ, µ2) is in the normal space (Drϕ(r, µ1), Drµ2(r, µ1))⊥ of Γ at the point (ϕ(r, µ1), µ2(r, µ1)). Hence, ϕ and

µ2 have to satisfy the additional equation:

Drϕ(r, µ1) ϕ+Drµ2(r, µ1) µ2 = 0. (5.36)

At first we solve Equation (5.35) together with Equation (5.36). Expanding gss w.r.t. (ϕ, µ2) we obtain

gss(ϕ+ 2l, µ1, µ2) = gss(ϕ, µ1, µ2) +Dϕgss(ϕ, µ1, µ2) ϕ+Dµ2

gss(µ2, µ1, µ2) µ2 +O(|(ϕ, µ2)|2).

Inserting this into (5.35) (and in the process exploiting (5.32)) yields

Dϕgss(ϕ, µ1, µ2) ϕ+Dµ2

gss(ϕ, µ1, µ2) µ2 +O(|(ϕ, µ2)|2)−∆sse2α

ss(µ1,µ2(r,µ1)+µ2)l(1 +O(e−αl)) = 0. (5.37)

Because of Hypothesis (H5.13), we may apply the Implicit Function Theorem to the system ((5.36), (5.37)), and

find a unique solution (ϕ∗, µ∗2)(l, r, µ1) for sufficiently small µ1 and |(ϕ, µ2)| and sufficiently large l. This solution

tends to zero uniformly in r and µ1 as l tends to infinity. Moreover, (ϕ∗, µ∗2)(l, r, µ1) is differentiable with respect to

l and Dl(ϕ∗, µ∗

2)(l, r, µ1) = O(e−αl). The latter can be seen by differentiating (5.37) and taking Corollary 5.3 into

consideration.

Next we consider (5.34), where we insert (ϕ∗, µ∗2). Due to Hypothesis (H5.6) (i), the resulting equation can be solved

in the same way as (5.18) in Section 5.2. We obtain the solution µ1 = µ1(l, r), and we find that both µ1(l, r) and

Dlµ1(l, r) are of order O(e−αl). The estimates for the derivatives follow from Corollary 5.3.

It remains to solve Equation (5.33), which can be written as

(ϕ(r, µ1(l, r)) + ϕ(l, r)− 2l) mod 2π = hc(µ1(l, r), µ2(r, µ1(l, r)) + µ2(l, r)), (5.38)

where (ϕ, µ2)(l, r) := (ϕ∗, µ∗2)(l, r, µ

∗1). To solve (5.38), we have to overcome similar obstacles as when solving Equation

(5.22). For that purpose we define

2l0(r) := ϕ(r, 0)− hc(0, µ2(r, 0)). (5.39)

Note that l0(r) : S1 → R is smooth. Now we set 2l = 2l0(r) + 2l + 2kπ, for some k ∈ N. Fixing k ∈ N, we define

µ1,k(l, r) := µ1(l0(r) + l + kπ, r), µ2,k(l, r) := µ2(l0(r) + l + kπ, r), ϕk(l, r) := ϕ(l0(r) + l + kπ, r).

Using this, we rewrite (5.38) as

ϕk(l, r) +O(µ1,k) +O(µ2,k) = 2l. (5.40)

This equation can be solved by means of the contraction principle. For that we note that the terms on the left-hand

side together with their derivatives are of order O(e−αk). Thus, for each fixed r and sufficiently large k, Equation (5.40)

5.4 Isolas 73

possesses a unique fixed point l∗k(r). Moreover, l∗k : S1 → R is smooth, and l∗k(r) is of order O(e−αk). All in all we

obtain the unique solutions

lk(r) = l0(r) + lk(r) + kπ, (5.41)

µ1,k(r) = µ1(lk(r), r),

µ2,k(r) = µ2(r, µ1,k(r)) + µ2(lk(r), r).

Obviously, Λk := (µ1,k(r), µ2,k(r)), r ∈ S1 are closed curves. Finally, with the transformation

Lk(r) = lk(r) +O(e−αlk(r)), (5.42)

see (5.16), we get

µ1,k(r) = µ1(Lk(r) +O(e−α lk(r)), r) = O(e−αk),

µ2,k(r) = µ2(r, µ1,k(r)) + µ2(Lk(r) +O(e−αlk(r)), r) = µ2(r, µ1,k(r)) +O(e

−αk).

Furthermore, we define the intervals Ik = [lk, lk] := Lk(S1). Due to Hypothesis (H5.14) and (5.39), the length of the

interval l0(S1) is less than π. Finally, since lk(r) = O(e

−αk), it follows with (5.41) and (5.42) that there is a d > 0

such that for sufficiently large k

lk+1 − lk > d. (5.43)

Hence, for sufficiently large k and k

Ik ∩ Ik = ∅.

It remains to show that the curves Λk are mutually disjoint: Assume that there exist k, k ∈ N, k > k and r, r ∈ S1

such that

(µ1, µ2) := (µ1,k(r), µ2,k(r)) = (µ1,k(r), µ2,k(r)).

Hence, huu(µ1, µ2) = huu(µ1,k(r), µ2,k(r)) = huu(µ1,k(r), µ2,k(r)), and from (5.34) we deduce

e−2αuu(µ1,µ2)lk(1 +O(e−αlk)) = e−2αuu(µ1,µ2)lk(1 +O(e−αlk)), (5.44)

where we exploited the fact that ∆uu depends only on µ1, µ2 and ϕ, together with Equation (5.33). Because of

lk(r)− lk(r) > d, see (5.43), we infer from (5.44) that

(1 +O(e−αlk)) = e−2αuu(µ1,µ2)[lk−lk](1 +O(e−αlk)) < e−2αuu(µ1,µ2)d + e−2αuu(µ1,µ2)dO(e−αlk). (5.45)

Taking the limit k, k →∞, we see that the left-hand side of (5.45) tends to 1. On the other hand the right-hand side

of (5.45) is close to e−2αuu(µ1,µ2)d < 1. This yields a contradiction, and for this reason the curves Λk and Λk cannot

intersect.

75

Chapter 6

The construction of Fenichel coordinates

In this chapter we prove Theorem 2.3, that is we establish Fenichel coordinates near the periodic orbit P under the

assumptions of Hypotheses (H2.7)–(H2.9). Fenichel coordinates are based on the work of Fenichel [18–21] and in the

framework of singular they have been established e.g. by Jones and Kopell [31], as well as, by Jones and Tin [32].

According to Hypothesis (H2.8), Equation (2.7) possesses a first integral, if the perturbation parameter λ = 0. Then

the periodic orbit P is (restricted to a level set of the first integral) hyperbolic. The construction of Fenichel coordinates

near a hyperbolic periodic orbit can be done similarly to the proofs of Jones and Kopell or Jones and Tin, respectively.

However, for λ 6= 0, the first integral does no longer exist and consequently the periodic orbit loses its hyperbolicity.

This changes qualitatively the dynamics near P and additional arguments are necessary to construct the Fenichel

coordinates. Here we carry out the analysis for

λ ≤ 0,

although similar arguments apply also for λ ≥ 0. The proof consist of the following two main parts : At first we derive

in Section 6.1 smooth foliations of the weak stable manifold W ss,eloc (P ) that corresponds to the Floquet multipliers

e2αssπ, e2α

eπ and the extended unstable manifold W e,uuloc (P ) that corresponds to the Floquet multipliers e2πα

e

, e2παuu

.

To construct these foliations we follow the analysis in [28] and [70]. After that we apply in Section 6.2 a series of

transformations that bring Equation (2.7) into the form of (2.8). In particular these transformations exploit the

foliations of W ss,eloc (P ) and W e,uu

loc (P ). The arguments in Section 6.2 go along to some extent with the considerations

in Jones and Kopell or Jones and Tin and we refer explicitly to the lecture notes [30], where those ideas are presented

very clearly.

6.1 A foliation of We,uuloc (P )

6.1.1 Preliminaries and main result

In this section we introduce some definitions and notations and state the main result regarding the existence of smooth

foliations of the manifolds W ss,eloc (P ) and W e,uu

loc (P ):

If M is a manifold and x ∈ M , we denote by TvM the tangent space of M at the point x and denote by TM the

tangent bundle of M , that is the set TM can be written as

TM =⋃

x∈M

TvM.

76 6 The construction of Fenichel coordinates

x

Fx

M

VM

U

F

Figure 6.1: A foliation F of V with base manifold U . The purple curve denotes the leaf Fx through x.

In accordance with Homburg [28] we use the notation foliation of a manifold. Let M be an m-dimensional manifold,

m ∈ N, and let V be an open subset of M . Denote by s, u ∈ N some numbers with s+ u = m. A foliation F of V is

a decomposition of V into s-dimensional disjoint, embedded submanifolds, called leaves, such that

(i) V is covered by C0-charts Υ : Ds ×Du →M , where Du ⊂ Ru and Ds ⊂ Rs denote unit balls in Ru and Rs.

(ii) If (xs, xu) ∈ Du and FΥ(xs,xu) is the leaf of F through Υ(xs, xu) ∈ V , then Υ(Ds × xu) ⊆ FΥ(xs,xu).

Observe that the restrictions of the charts Υ : 0 × Du → M form (a covering of) an u-dimensional, embedded

submanifold U of M . We call U the base manifold of the Foliation F . Figure 6.1 provides a visualisation of this

definition.

We call a foliation F of V with base manifold U to be Ck-smooth, if the charts Υ are Ck-smooth. Furthermore we

say that the foliation F is flow invariant (w.r.t. the flow ϕt(·) on U), if

ϕt(Fx) ∩ V ⊂ Fϕt(x), ∀t ∈ R, x ∈ V.

According to [28], the Ck-smoothness of the mapping x → TvFx implies the Ck-smoothness of the foliation F . For

more information about foliations we refer to [18,27].

Remark 6.1. Note that we use here a ”rather restrictive” definition of the term foliation, since we claim the leaves

to be embedded submanifolds. Other definitions [27,47] demand merely the leaves to be immersed submanifolds.

Next we recall Hypothesis (H2.9) and the definition of W ssloc(P ) and Wuu

loc (P ) as the (strong) stable and (strong)

unstable manifolds of P corresponding to the Floquet multipliers e2παss

and e2παuu

, respectively. These manifolds

depend smoothly on the parameters µ, λ, what we drop from our notation.

Corresponding to the Floquet multipliers e2παss

, e2παe

and e2παe

, e2παuu

we introduce furthermore the manifolds

W ss,eloc (P ) and W e,uu

loc (P ), respectively. Since e2παe

≤ 1, it is well known that W ss,eloc (P ) exists and is smooth. The

existence and smoothness of the manifold W e,uuloc (P ) is provided by [65, Theorem 5.19], which reads in our notation:

6.1 A foliation of W e,uuloc (P ) 77

Theorem 6.1 (Extended Manifolds, [65, Theorem 5.19]). Let m ∈ R and f = f(λ) be a family of vector fields in

Rm that possesses a 2π periodic orbit P for each λ. Assume that P has n Floquet exponents greater or equal than

αe ≤ 0 and that the other negative Floquet exponents of P are smaller or equal than αss < αe ≤ 0. Define q as the

largest integer such that |αe|q < |αss| and q ≤ k. Then for any λ the periodic orbit has an extended unstable (n+ 1)-

dimensional invariant Cq-smooth manifold W e,uuloc (P ). At each point of P this manifold is tangent to the eigensubspace

corresponding to the n Floquet exponents greater equal than αe. Furthermore, W e,uuloc (P ) contains the strongly unstable

manifold Wuuloc (P ). W e,uu

loc (P ) is not unique, but any two of them have the same tangent at any point of Wuuloc (P ).

Moreover, if f is Ck with respect to λ, then the manifold W e,uuloc (P ) is also Cq with respect to λ.

The proof of Theorem 6.1 is based on [65, Theorem 5.13], which has been proved in detail in [65]. The proof of

Theorem 6.1 follows the idea of the Perron method to prove the existence of corresponding invariant manifolds. To

this end one applies a cut-off function on the higher order terms of the given vector field f . This yields a vector field

f , which coincides with f locally around P and the higher order terms of f vanish outside a certain neighbourhood of

P . Consequently f is complete. Thus f has unique invariant manifolds.

Finally we define the 2-dimensional manifold

W eloc(P ) :=W ss,e

loc (P ) ∩W e,uuloc (P ).

Observe that if λ = 0, then W eloc(P ) the unique centre manifold, consisting of the one parameter family of periodic

orbits, which is due to the presence of the first integral, cf. Hypothesis (H2.8).

With these definitions we formulate the main result of this section:

Theorem 6.2. Let λ ≤ 0. The manifolds W ss,eloc (P ) and W e,uu

loc (P ) have smooth, flow invariant foliations Fss, Fuu

with base manifold W eloc(P ):

W ss,eloc (P ) =

⋃

x∈W eloc

(P )

Fssx , W e,uuloc (P ) =

⋃

x∈W eloc

(P )

Fuux .

The foliations Fss, Fuu consists of 1-dimensional strong stable and strong unstable leaves, respectively. Fss, Fuu

depend smoothly on µ, λ and for λ = 0 these foliations are R-images of each other: RFuux = FssRx, x ∈ W eloc(P ).

Moreover the leaves with base points in P form the strong stable and strong unstable manifolds of P :

W ssloc(P ) =

⋃

v∈P

Fssv , Wuuloc (P ) =

⋃

v∈P

Fuuv .

Finally, for all v ∈ P , it holds true that TγFssv = F ssv and TγF

uuv = Fuuv .

As already mentioned the proof of Theorem 6.2 is based on ideas of Homburg [28] and Vanderbauwhede and Tak-

ens [70]. In his work Homburg [28] has established smooth foliations in the vicinity of a homoclinic orbit. In [70]

Vanderbauwhede and Takens have explained the construction of foliations of centre, stable and centre-unstable mani-

folds of fixed points.


Before we start to construct the foliations, we consider the following result that we exploit repeatedly in this and the

following sections:

Lemma 6.1. Let f be a vector field on a manifold M and assume that f is reversible with respect to the reverser

R. Let T : M → M be a smooth coordinate transformation and let f be the push forward of f , i.e. f(T (x)) =

DT (x) f(x), x ∈M . If R is the linear involution defined by R = T R T−1, then f is reversible with respect to R:

R f = −f R.

Proof of Lemma 6.1. Let x be a vector in the corresponding coordinates of f . From R T = T R it follows that

RDT (x) = DT (Rx)R. Let v = T (x), then

Rf(v) = RDT (x)f(x) = DT (Rx)Rf(x) = −DT (Rx) f(Rx) = f(T (Rx)) = f(Rv).

6.1.2 A moving coordinate system near P

In this section we prove the following lemma that introduces local coordinates near the periodic orbit P . These

coordinates will be the basis for the constructions in the next section.

Lemma 6.2. Let the vector field of (2.7) be Ck smooth. Locally near the periodic orbit P there exist coordinates

(vc, ve, vss, vuu) ∈ S1 × R3 such that

(i) P = ve = 0, vss = 0, vuu = 0,

(ii) The sets vss = 0 and vuu = 0 are the extended stable and unstable manifolds W e,uuloc (P ) and W ss,e

loc (P ).

(iii) The vector field (2.7) has in the transformed coordinates the form:

f(vc, ve, vss, vuu) =

1 +O(ve) +O(vss) +O(vuu)

O(ve) +O(vss) +O(vuu)

(αss +Ass(vc, ve, vss, vuu))vss

(αuu +Auu(vc, ve, vss, vuu))vuu

.

The transformed vector field f depends Ck-smoothly on v and the parameters µ, λ. Furthermore Ass(vc, 0, 0, 0) = 0

and Auu(vc, 0, 0, 0) = 0 for all vc ∈ S1.

(iv) If λ = 0, then the vector field is reversible in the (vc, ve, vss, vuu)-coordinates and the reverser R given by

Hypothesis (H2.7) is described by the linear involution (vc, ve, vss, vuu) 7→ (−vc, ve, vuu, vss).

Moreover, if |λ| is sufficiently small and hence αe is sufficiently close to zero, then locally around P the transformation

into these coordinates is a Ck-smooth mapping.

We refer to [25, VI.1] and [65, Theorem 3.11], where also a moving coordinate system near a periodic is established.

These works are closely related to our considerations here and use similar arguments.


Proof of Lemma 6.2 To construct the transformation of Lemma 6.2 we choose at any point γt of P vectors

ekt , k = ss, e, uu that depend smoothly on γt and are invariant under the linearised flow Dϕτ (γt)τ∈R. Together with

the vector field direction f(γt) these vectors form a moving coordinate system around P .

A common choice for ekt , k = ss, e, uu are normalised vectors that lie within the eigenspaces of the monodromy matrices

Dϕ2π(γt) corresponding to the Floquet multipliers e2παe

, e2παss

, e2παuu

, see [25, 65]. If λ = 0, then the periodic orbit

possesses the double, semi-simple Floquet multiplier 1. Hence the eigenspace TγtWe(P ) of Dϕ2π(γt) that corresponds

to the eigenvalue e2παe

= 1 is 2-dimensional. Thus we need to specify eet within TγtW

e(P ). Denote by < (·), (·) > the

Euclidean inner product. We define the R invariant inner product

< (·), (·) >R :=< (·), (·) > + < R(·), R(·) >,

which is well defined also for nonzero λ. To verify that < (·), (·) >R is an inner product, one exploits the linearity of

R. Now define normalised vectors eet ∈ TγtW

e(P ) such that eet are orthogonal to f(γt) w.r.t. < (·), (·) >R. Thereby

we choose eet such that the vectors depend continuously on t ∈ [0, 2π]. Observe that if λ 6= 0 (i.e. e2πα

e

6= 1), then eet

are in general not eigenvectors of the monodromy matrix Dϕ2π(γt).

Next we choose normalised eigenvectors, ess0 and euu0 , of Dϕ2π(γ0) that corresponds to the eigenvalue e2πα

ss

and e2παuu

,

respectively. We make this choice so that ess0 and ess0 are smooth with respect to µ, λ. Moreover we demand that

Reuu0 = ess0 , λ = 0.

This is possible since for λ = 0, it holds true that γ0 = γ(0, µ, 0) ∈ Fix(R) and RDϕ2π(γ0) = Dϕ−2π(γ0)R, as well as,

αss(µ, 0) = −αuu(µ, 0). Finally we define

esst := e−α

sstDϕt(γ0)ess0 and e

uut := e−α

uutDϕt(γ0)euu0 , t ∈ R. (6.1)

It follows that if λ = 0, then Reuut = esst . By definition e

uut , esst depend smoothly on µ, λ and t. Further we introduce

Pγτ

γt

f(γt)

f(γτ )

euuγt

euuγτ

eeγt

eeγτ

W eloc(P )

Figure 6.2: The moving coordinate system on P restricted to W e,uuloc (P ).

F eγt := spaneet, F ssγt := spanesst , Fuuγt := spaneuut and St := γt + F eγt ⊕ Fssγt ⊕ F

uuγt . (6.2)

The sections St form a tubular neighbourhood of P [47]. Let U be a sufficiently small neighbourhood of P and x ∈ U .


Thus for any x ∈ U there exists a well-defined t = τ(x) ∈ [0, 2π) such that x belongs to the section Sτ(x):

x = γτ(x) + xe(x) · eeτ(x) + xss(x) · essτ(x) + xuu(x) · euuτ(x), xe, xss, xuu : R4 → R. (6.3)

It is well known that the mapping

T0 : x 7→ (τ, xe, xss, xuu) := (τ(x), xe(x), xss(x), xuu(x))

is a transformation. In each section Sτ the trace of the manifoldsW ss,eloc (P ) andW e,uu

loc (P ) are locally near P the graph

of a Ck-smooth function

Ψss,eτ : F eτ ⊕ Fssτ → Fuuτ ,

Ψe,uuτ : F eτ ⊕ Fuuτ → F ssτ ,

respectively. For the Ck smoothness we recall that W ss,eloc (P ) and W e,uu

loc (P ) are Ck, see Theorem 6.1. Moreover,

Ψss,eτ (0) = 0 and Ψe,uuτ (0) = 0 and at 0 the functions Ψss,eτ and Ψe,uuτ are tangent to F eτ ⊕ F ssτ and F eτ ⊕ Fuuτ ,

respectively. In accordance with the transformation (6.3) we identify F eτ ⊕ Fssτ and F eτ ⊕ F

uuτ with R2 and write

Ψss,eτ (xe, xss) := Ψss,eτ (xe · eeτ + xss · essτ ),

Ψe,uuτ (xe, xuu) := Ψe,uuτ (xe · eeτ + xuu · euuτ ).

Consequently,

Ψss,eτ (0, 0) = 0, Ψe,uuτ (0, 0) = 0 and D(xer,x

ssr )Ψ

ss,eτ (0, 0) = 0, D(xe

r,xuur )Ψ

e,uuτ (0, 0) = 0, ∀ τ ∈ S1. (6.4)

And therefore

DτΨss,eτ (0, 0) = 0 and DτΨ

e,uuτ (0, 0) = 0, ∀ τ ∈ [0, 2π). (6.5)

Next we consider the following transformation:

T1 : (τ, xe, xss, xuu) 7→ (vc, ve, vss, vuu) :=(

τ, xe, xss −Ψe,uuτ (xe, xuu), xuu −Ψss,eτ (xe, xss))

that brings the vector field f into the from given in Lemma 6.2. To show that T1 is indeed a transformation we exploit

the Inverse Function Theorem. From (6.4) and (6.5) it follows that DT1(τ, 0, 0, 0) = I. Hence DT1 is invertible. By

the Inverse Function Theorem T1 possesses an inverse locally around any point (τ, 0, 0, 0) ∈ P . It remains to prove

that T1 is also globally invertible:

Assume that there are two points x = (τx, xe, xss, xuu) and y = (τy, y

e, yss, yuu) such that T1(x) = T1(y). By the

definition of T1 it follows that τx = τy. Hence if xe, xss, xuu and ye, yss, yuu are sufficiently close to zero, then x and y

belong to a neighbourhood of the point (τx, 0, 0, 0). Due to the local invertibility, T1 is invertible in that neighbourhood.

Finally we observe that the Inverse Function Theorem implies that T1 is Ck, since Ψss,e and Ψe,uu are Ck.


We observe that by the construction of T1 the assertions (i) and (ii) of the lemma are satisfied. Next we prove

statement (iii). First we derive the structure of f after the transformation T0. We show that

f(τ, xe, xss, xuu) =(

1, 0, αss xss, αuu xuu)T

+O(xe) +O((xss + xuu)2). (6.6)

We start with a Taylor expansion of f with respect to xe,

f(τ, xe, xss, xuu) = f(τ, 0, xss, xuu) +O(xe).

Thus we need to calculate f(τ, 0, xss, xuu). By the definition of T0, the point (τ, 0, xss, xuu) is given in cartesian

coordinates by γτ + xss essτ + xuu euuτ . Hence

f(τ, 0, xss, xuu) = DT0(γτ + xssessτ + xuueuuτ ) f(γτ + xssessτ + xuueuuτ )

= DT0(γτ + xssessτ + xuueuuτ )[

f(γτ ) + xssDf(γτ )essτ + xuuDf(γτ )e

uuτ +O

(

(xssessτ + xuueuuτ )2)]

.(6.7)

Next we calculate the expression DT0(γτ + xssessτ + xuueuuτ )[


uuτ

]

. According to

the definition of essτ , euuτ , see (6.1),

(τ, 0, xss, xuu) = T0(

γτ + xssessτ + xuueuuτ)

= T0(

γτ + xsse−αssτ Dϕτ (γ0)e

ss0 + xuue−α

uuτ Dϕτ (γ0)euu0

)

. (6.8)

Hence differentiating Equation (6.8) with respect to τ gives

(1, 0, 0, 0) = DT0(γτ + xss essτ + xuu euuτ ) ·[

f(γτ ) + xsse−αssτ Df(γτ )Dϕ

τ (γ0)ess0 + xuue−α

uuτ Df(γτ )Dϕτ (γ0)e

u0

]

−DT0(γτ + xss essτ + xuu euuτ ) ·[

αssxsse−αssτ Dϕτ (γ0)e

ss0 + αuuxuue−α

uuτ Dϕτ (γ0)eu0

]

= DT0(γτ + xss essτ + xuu euuτ ) ·[


uuτ

]

−DT0(γτ + xss essτ + xuu euuτ ) ·[

αssxss essτ + αuuxuu euuτ]

. (6.9)

Here we made use of the fact that Dϕτ (γ0) solves the initial value problem x = Df(γτ )x, x(0) = id and of (6.1).

Further, differentiating (6.8) with respect to xss and xuu gives

(0, 0, 1, 0) = DT0(γτ + xss essτ + xuu euuτ ) essτ and (0, 0, 0, 1) = DT0(γτ + xss essτ + xuu euuτ ) euuτ .

With that and (6.9) we get

(1, 0, αssxss, αuuxuu) = DT0(γτ + xss essτ + xuu euuτ ) ·[


uuτ

]

. (6.10)

Plugging (6.10) into (6.7) yields the representation of f given in (6.6)

f(τ, 0, xss, xuu) = (1, 0, αss · xss, αuu · xuu)T +O(

(xss + xuu)2)

.

To prove (iii) it remains to consider the effect of the transformation T1 on the structure of the vector field f . At first

we observe that under T1 the structure of ’original’ vector field (after applying T0) is preserved. This is due to the fact

that the base vectors essτ , euuτ lie also after applying T1 within the tangent spaces of W ss

loc(P ) and Wuuloc (P ), respectively.


Note that (6.6) already provides the form of f c and fe. To obtain the desired form of fss and fuu we exploit that by

the construction of T1

W ss,eloc (P ) = vuu = 0 and W e,uu

loc (P ) = vss = 0.

By the flow invariance of those manifolds we infer fss(τ, ve, 0, vuu) = 0 and fuu(τ, ve, vss, 0) = 0. Consequently

fss(τ, ve, vss, vuu) = Ass(τ, ve, vss, vuu)vss, fuu(τ, ve, vss, vuu) = Auu(τ, ve, vss, vuu)vuu,

for some smooth functions Ass and Auu. Together with (6.6) we may write Ass = αss + Ass and Auu = αuu + Auu.

This yields the structure of the vector field.

Finally we prove (iv) of the lemma. Let λ = 0. At first we claim that the transformation T0 is reversible with respect

to the linear involution

R0(τ, xe, xss, xuu) := (−τ, xe, xuu, xss),

i.e. R0 T0 = T0 R.

To this end we show that

(i) Reuut = ess−t, Resst = e

uu−t and (ii) Reet = e

e−t, ∀γt ∈ P. (6.11)

Recall that γ0 ∈ Fix (R). To show (6.11) (i) we note at first that ess0 , e

uu0 are eigenvectors of Dϕ2π(γ0). Moreover

Dϕt(γ0) is reversible, i.e. RDϕt(γ0) = Dϕ−t(γ0)R = (Dϕt(γ0))−1R. Hence the eigenvector e

k0 corresponding to the

eigenvalue e2παk

, k = ss, e, uu satisfies

e2παk

R ek0 = RDϕ2π(γt) e

k0 = (Dϕ2π(γt))

−1R ek0 and consequently Dϕ2π(γt)R e

k0 = e−2παk

R ek0 ,

which shows that Rek0 is an eigenvector of Dϕ2π(γ0) to the eigenvalue e−2παk

. Combining this with Rγ0 = γ0 and

αss(µ, 0) = −αuu(µ, 0) yields (6.11) (i) for t = 0. For arbitrarily γt ∈ P we use (6.1) to calculate that

Reuut = e−αuutRDϕt(γ0)e

uu0 = eα

sstDϕ−t(γ0)Reuu0 = eα

sstDϕ−t(γ0)ess0 = e

ss−t.

Note that the last equality is due to the symmetry of P with respect to Fix (R). Altogether this proves (6.11) (i).

Concerning (6.11) (ii) we note that similarly to (6.11) (i) one can show that

RTγtWeloc(P ) = Tγ−t

W eloc(P ), ∀γt ∈ P. (6.12)

At first we consider (6.11) (ii) for t = 0,

Ree0 = ee0,

i.e. ee0 ∈ Fix(R). By construction spanf(γ0), e

e0 is R invariant, and since f(γ0)⊥ e

e0 also spanee0 is R invariant.

Consequently ee0 ∈ Fix(R) or e

e0 ∈ Fix(−R). If ee0 ∈ Fix(−R), then Fix(−R) = spanf(γ0), e

e0. But this contradicts

ess0 − e

uu0 ∈ Fix(−R).


To see that (6.11) (ii) is satisfied for all γt ∈ P we calculate

0 = −⟨

f(γt), eet

⟩

R=

⟨

f(γ−t), R eet

⟩

R.

With (6.12) this gives eet ∈ spanee−t and the continuity of eet with respect to t and Ree0 = ee0 imply (6.11) (ii).

Finally let x ∈ R4 be a point near the periodic orbit and let T0(x) = (τ, xe, xss, xuu), then (6.11) and the symmetry

of P yield

R0(τ, xe, xss, xuu) := (−τ, xe, xuu, xss) = T0(γ−τ + xeee−τ + xuuess−τ + xsseuu−τ ) = T0(Rx).

In other words R0 T0 = T0 R.

Next we discuss the reversibility of the transformation T1. Recall that for λ = 0 the manifoldsW ss,eloc (P ) andW e,uu

loc (P )

are uniquely defined centre-stable and centre-unstable manifolds, respectively. Hence

RW ss,eloc (P ) =W e,uu

loc (P ).

Moreover we claim that in case λ = 0

Ψss,eτ (xe, xss) = Ψe,uu−τ (xe, xss). (6.13)

To see (6.13) we consider the following diagram, see also Figure 6.3,

(τ, xe, xss, 0) ∈ F ssτR0−−−−−→ R0(τ, x

e, xss, 0) = (−τ, xe, 0, xss) ∈ Fuu−τ

−→

Ψe,uu−τ

R0(−τ, xe,Ψe,uu−τ (xe, xss), xss)

= (τ, xe, xss,Ψe,uu−τ (xe, xss)) ∈W ss,eloc (P )

R0←−−−−− (−τ, xe,Ψe,uu−τ (xe, xss), xss) ∈W e,uuloc (P ).

Hence the construction yields a map

(τ, xe, xss, 0) ∈ F ssτ 7→ (τ, xe, xss,Ψe,uu−τ (xe, xss)) ∈W ss,eloc (P ).

And thus the uniqueness of Ψss,eτ yields (6.13).

Now we finish the proof of statement (iv). In the (vc, ve, vss, vuu) coordinates the reverser R acts as R = T1 R0 T−11 .

We prove that R(vc, ve, vss, vuu) = (−vc, ve, vuu, vss) by exploiting (6.13):

R(vc, ve, vss, vuu) = T1 R0(τ, xe, xss, xuu) = (−τ, xe, xuu −Ψss,eτ (xe, xss), xss −Ψe,uuτ (xe, xuu))

= (−τ, xe, xuu −Ψe,uu−τ (xe, xss), xss −Ψss,e−τ (xe, xuu))

= (−vc, ve, vss, vuu).


F ss(τ,0,0,0)

(τ, 0, 0, 0)

W ss,eloc (P )

Fuu(−τ,0,0,0)

(−τ, 0, 0, 0)

R0

R0

Ψe,uu−τ

(τ, xe, xss, 0)

(−τ, xe, xss, 0)

W e,uuloc (P )

R0 Ψe,uu−τ R0

P

P

Figure 6.3: The map R0 Ψe,uu−τ R0.

6.1.3 Foliations of We,uuloc (P ) and W

ss,eloc (P )

In this section we derive the foliations of W e,uuloc (P ) and W ss,e

loc (P ) with the base manifold W eloc(P ). To this end

we consider a certain vector bundle on the manifold W e,uuloc (P ). On this bundle we define an (extended) flow that

corresponds to the flow that is induced by the vector field f on W e,uuloc (P ) and that encodes the (linearised) structure

of f . We show that this extended flow possesses a periodic orbit that corresponds to the periodic orbit P . Then

we prove that the extended periodic orbit possesses an extended unstable manifold. By the means of this extended

unstable manifold we derive finally the foliation of W e,uuloc (P ).

At first we recall some of the definitions of the last section and introduce some further conventions and definitions that

we need in the following. We recall that λ ≤ 0 and that [65, Theorem 5.19] provides the existence and smoothness

of the (non unique) extended unstable manifold W e,uuloc (P ). Moreover, since αe tends to zero as λ tends to zero, we

may assume the manifold W e,uuloc (P ) to be as smooth as the vector field f (for sufficiently small |λ|). Further we recall

the notations Fe/ss/uut := spane

e/ss/uuγt introduced in (6.2). In what follows we use the coordinates introduced in

Lemma 6.2, where we drop the vss-component from the notation. That means that we denote points in W e,uuloc (P ) by

v = (vc, ve, vuu).

Similarly we define [ϕσ(v)]c := ([ϕσ(v)]c, 0, 0), v ∈W e,uuloc (P ).

By means of the Grassmanian manifold, that consists of all 1-dimensional linear subspaces of TvWe,uuloc (P ) [47, Example

1.24] we introduce the Grassmanian bundle G1(TWe,uuloc (P )), cf. [28, 70]:

G1(TWe,uuloc (P )) :=

⋃

v∈W e,uu

loc(P )

(

v,G1(TvWe,uuloc (P ))

)

.


Dϕσ(xγ)

φσuu

ϕσ(xγ) [ϕσ(x)]γ

x

xγ

eexγ

f(xγ)

f([ϕσ(x)]γ)

ee[ϕσ(x)]γ ϕσ(x)

PSτ(x)

PSτ([ϕσ(x)]γ)

P

Aγ

Bγ

f(ϕσ(xγ))eeϕσ(xγ)

Figure 6.4: The flow on the vector bundle Ge,uu

Further information on the Grassmanian bundle can be found in [27]. The object of our further considerations is the

following subbundle of Grassmanian bundle:

Ge,uu :=⋃

v∈W e,uu

loc(P )

(

v,L(Fuuvc , spanf(γ(vc)) ⊕ F evc)

)

,

where L(Fuuvc , spanf(γ(vc))⊕F evc) is the space of linear mappings from Fuuvc to spanf(γ(vc))⊕F evc . On this bundle

we define φσuu(·, ·)σ∈R:

φσuu : Ge,uu → Ge,uu,

φσuu(v,A) := (ϕσ(v),B(v,A)), (6.14)

where B = B(v,A) ∈ L(Fuu[ϕσ(v)]c , spanf([ϕσ(v)]c)⊕F e[ϕσ(v)]c) such that graphB = Dϕσ(v)(graphA). For a graphical

illustration see Figure 6.4.

Lemma 6.3. φσuu(·, ·)σ∈R is a flow on the bundle Ge,uu, which possesses a 2π-periodic orbit

P := (v, 0γ(vc)) | v ∈ P,

where 0γ(vc) denotes the zero map in L(Fuuvc , spanf(v) ⊕ Fevc).

Proof of Lemma 6.3. We note that due to (6.1), φσuu is well defined. We calculate

(φσ1uu φ

σ2uu)(v,A) = φσ1

uu(φσ2uu(v,A)) = φσ1

uu(ϕσ2(v),B) = (ϕσ1+σ2(v), B),

where

graph B = Dϕσ1(ϕσ2(v))(graphB) = Dϕσ1(ϕσ2(v))(

Dϕσ2(v)(graphA))

=d

d v(ϕσ1(ϕσ2(v))(graphA)

= Dϕσ1+σ2(v)(graphA).


Moreover φ0uu(v,A) = (v,A).

It remains to prove that P is a periodic orbit. For this we observe that graph 0γ(vc) = Fuuvc , for some γ ∈ P . Inspecting

(6.1), we see that Dϕ2π(γ)Fuuvc = Fuuvc and thus Dϕ2π(γ)(graph 0γ(vc)) = graph 0γ(vc).

Lemma 6.4. The periodic orbit P possesses the Floquet multipliers e2παe

, 1, e2παuu

(these are the Floquet multipliers

of P ) and e2π(αe−αuu) and e−2παuu

.

We want to note again that αe/uu = αe/uu(µ, λ) and that in particular e2παe(µ,0) = 1.

Proof of Lemma 6.4. The Floquet multipliers are the eigenvalues of the monodromy matrix:

d

d(v,A)φ2πuu(v,A)|v=γ0,A=0γ(0)

=

(

Dϕ2π(γ0) 0

DvB(γ(0), 0γ(0)) DAB(γ(0), 0γ(0))

)

.

The eigenvalues of Dϕ2π(γ) are the Floquet multipliers of P . In order to compute the eigenvalues of DAB(γ(0), 0γ(0)),

let graphA =

z(

euu0 + a

c · f(γ0) + ae · ee0

)

| z ∈ R

.

With that we may write

B(γ(0),A) = e−2παuu

Dϕ2π(γ(0))

(

ac

ae

)

,

where Dϕ2π(γ(0)) denotes the corresponding matrix representation of Dϕ2π(γ(0)) restricted to Tγ(0)Weloc(P ). Hence

DAB(γ(0),A) = e−2παuu

Dϕ2π(γ(0))

(

1 0

0 1

)

= e−2παuu

Dϕ2π(γ(0)).

Since Dϕ2π(γ(0)) possesses the eigenvalues 1 and e2παe

, we find that e−2παuu

and e2π(αe−αuu) are the eigenvalues of

DAB(γ(0), 0γ(0)).

According to Lemma 6.4 and Theorem 6.1 the periodic orbit P possesses a 3-dimensional extended unstable manifold,

We,uu(P) that is related to the Floquet multipliers e2παe

, 1, e2παuu

. Due to Theorem 6.1 this manifold is Ck-smooth

in v,A and µ, λ, for sufficiently small |λ|.

Lemma 6.5. The manifold W e,uuloc (P ) has a foliation Fuu in 1-dimensional leaves with base manifold W e

loc(P ). More-

over the leaves with base points in P form the unstable manifold of P :

Wuuloc (P ) =

⋃

v∈P

Fuuv

and for all v ∈ P , it holds TγFuuv = Fuuv .

Proof of Lemma 6.5. Recall that according to Theorem 6.1 the manifold We,uu(P) is not unique. However,

once We,uu(P) is chosen, for any v ∈ W e,uuloc (P ) there is a unique Av ∈ L(F

uuvc , spanf(γ(v

c)) ⊕ F evc) such that

(v,Av) ∈ We,uu(P). Similarly as in the proof of Lemma 6.4 we identify Av with the mapping

z 7→ (acv z, aev z), v ∈W e,uu

loc (P ).

Since We,uu(P) is smooth, both acv and a

ev depend smoothly on v and µ, λ. Hence in the (vc, ve, vuu) coordinates,


(acv, aev, 1) ∈ R3 defines a smooth vector field, Fe,uu, on W e,uu

loc (P ). Due to the Picard/Lindelof Theorem, this vector

field is integrable. We define Fuuv as the Fe,uu orbit through v.

Obviously the orbit Fuuv is a 1-dimensional manifold that depends smoothly on v. Since Fe,uu is smooth in µ, λ, any

orbit Fuuv is smooth in these variables. Furthermore, by the definition of the flow φσuu(·, ·)σ∈R, (6.14), it follows that

Fuuv is ϕσ-flow invariant.

In order to reveal W eloc(P ) as base manifold we simply mention that for all v ∈ W e

loc(P ) the leaf Fuuv and W eloc(P )

intersect transversally in v.

For the assertion regarding the tangent space TγFuuv , v ∈ P , we exploit that P ⊂ We,uu

loc (P) and that P = (P, 0γ(vc)),

see Lemma 6.4. Finally we recall that 0γ(vc) corresponds to euuγ(vc) + 0 · eγ(vc) + 0 · f(γ(vc)).

For the remaining statement of the lemma, we consider the 2-dimensional manifold

M :=⋃

v∈P

Fuuv .

Since the leaves Fuuv are invariant with respect to the flow ϕt(·)t∈R, the manifold M is flow invariant. According

to the generalised Hartman/Grobman Theorem there are exactly two flow invariant, 2-dimensional manifolds near

P , namely W eloc(P ) and Wuu

loc (P ). Therefore the construction of M implies that M = Wuuloc (P ). Furthermore, by

definition, Fuu restricted to P is a foliation of M =Wuuloc (P ).

Remark 6.2. We deduced the integrability of the vector field Fe,uux by exploiting the Picard/Lindelof Theorem. Note

that this is only applicable, since dim(Fe,uux ) = 1. For higher dimensional leaves one can follow the approach of

Vanderbauwehde/Takens [70] and Homburg [28] respectively.

The existence of a smooth, flow invariant foliation Fss of W ss,eloc (P ) with base manifold W e

loc(P ) is well known, see

e.g. [65, Theorem 5.20]. Moreover there is a unique foliation of the strong stable manifold W ssloc(P ). Take any point

v ∈W ssloc(P )\P . According to the invariance ofW ss

loc(P ), the leaf Fssx ⊂W

ssloc(P ). In particular for any v ∈W ss

loc(P )\P

there exists a v ∈ P such that Fssγ ⊂Wssloc(P ). By the uniqueness of the foliation of W ss

loc(P ) it follows that

W ssloc(P ) =

⋃

v∈P

Fssγ .

This finishes the proof of Theorem 6.2, except for the assertions concerning the reversibility.


6.1.4 Reversibility and dependency on parameters

In our explanations the existence of the manifolds We,uuloc (P) and Wss,e

loc (P) is due to Theorem 6.1. According to that

theoremWe,uuloc (P) andWss,e

loc (P) are not unique. ButWe,uuloc (P) andWss,e

loc (P) can be chosen such that these manifolds

depend smoothly on the parameters µ, λ. This is also true for the ”nonlocal” parameter area Jµ.

Moreover, we claim that these manifolds can be chosen such that for λ = 0 the corresponding foliations Fuu and Fss

are R images of each other, i.e. RFuux = FssRx, x ∈ Weloc(P ). In this respect we recall that for λ = 0 the manifold

W eloc(P ) is unique; it is the manifold consisting of a one parameter family of periodic orbits. Due to the uniqueness

we find

RW eloc(P ) =W e

loc(P ).

Lemma 6.6. There are foliations Fss and Fuu depending smoothly on the parameters µ, λ that satisfy the assertions

of Lemma 6.5 such that for λ = 0

RFuux = FssRx, x ∈W eloc(P ).

Proof of Lemma 6.6. Let f = f(·, µ, λ) be a vector field evolved out of f by the cut-off function χf , as described in

our comments regarding Theorem 6.1. If χf is R-invariant, then the vector field f(·, µ, 0) is R-reversible and for λ = 0

the unique manifolds Wss,eloc (P) and We,uu

loc (P) are R-images of each other. Based on these manifolds we consider the

flows φσssσ∈R and φσuuσ∈R, which are defined by means of the flow f . Further we introduce a ”reverser”R by

R(x,A) :=(

Rx,BR)

, where graphBR = R graphA.

We find that

Rφσuu(x,A) = φ−σss(

R(x,A))

.

This yields that the unique manifolds Wss,eloc (P) and We,uu

loc (P) are R-images of each other:

RWe,uuloc (P;µ, 0) =Wss,e

loc (P;µ, 0).

Denote Fss,e the vector field that is defined analogously to Fuu. Then the above line yields that

RFe,uu(x) = −Fss,e(Rx).

This finally gives the lemma.

Combining Lemma 6.5 and Lemma 6.6 yields Theorem 6.2.

6.2 The vector field in the Fenichel coordinates 89

6.2 The vector field in the Fenichel coordinates

In this section we finish the proof of Theorem 2.3. We construct a series of Ck-smooth transformations that convert

the vector field given in (2.7) into the form stated in (2.8). Further we show that the reverser R acts as stated in

(2.9). The transformations we are going to establish exploit among others the Ck-smooth foliations of W ss,eloc (P ) and

W e,uuloc (P ) that were constructed in the previous Section. The following observations show that these transformations

do not alter the Floquet multipliers of P :

Lemma 6.7. Let T be a transformation, which is defined locally near P , with T|P = id. Further let ϕt(·)t∈R denote

the flow of the transformed vector field

f(Tv) = DT (v)f(v).

Then the Floquet multipliers of P with respect to f coincide with the Floquet multipliers of P with respect to f .

Proof of Lemma 6.7. Since T ϕt = ϕt T , it follows DT (γ)Dϕ2π(γ) = Dϕ2π(γ)DT (γ) for all γ ∈ P .

The construction of the Fenichel coordinates is similar to the work by Jones [30, Chapter 3], Jones and Tin [32]. There

a vector field with a similar structure is has been gained in the context of a singular perturbed system. Recall that

in [25, VI.1] and [65, Theorem 3.11] a moving coordinate system near a periodic is established. We remark that both

of those works do not take the Floquet multipliers of the periodic orbit into account, when constructing the local

coordinates. Consequently the ”linear rates” of the vector fields are not worked out there.

In what follows we suppress the dependency on the parameters µ, λ in our notation.

The first transformation we consider straightens the leaves of the stable and unstable foliations Fuu and Fss, see

Figure 6.5. The effect of this is that the leaves of the foliations Fuu and Fss in the new coordinates are described by

the following sets (vc, ve, s, 0) | s ∈ Iδ and (vc, ve, 0, u) |u ∈ Iδ. Now we consider the addressed transformation in

W e,uuloc (P )

P

Fuu

Fuu(τ,0,0,0)

(τ, xe, xss, xuu)

T2

W e,uuloc (P )

P

Fuu

Fuu(xc,0,0,0)

(xc, xe, xss, xuu)

Figure 6.5: Straightening the leaves of the foliation Fuu.

more detail. The following observations help to define this transformation. Let Fuu(vc,ve,0,0) be the leaf of the foliation

Fuu through the base point (vcb , veb , 0, 0). Observe that (vcb , v

eb , 0, 0) ∈ W e

loc(P ) = W ss,eloc (P ) ∩W e,uu

loc (P ), according

to Lemma 6.2 (ii). The bundle projection Πuu : W e,uuloc (P ) ⊂ R4 → R2 maps v ∈ Fuu(vc,ve,0,0) onto (vc, ve). Of

course the bundle projection is Ck. We define a mapping (that we also call) Πuu on R4 by Πuu(vc, ve, vss, vuu) :=

Πuu(vc, ve, 0, vuu). Note that Πuu is Ck smooth.


Further we observe that by the definition of Fuu there are functions huu(vc,ve) : R→ R2 such that

Fuu(vc,ve,0,0) =

(huu(vc,ve)(vuu), 0, vuu) | vuu ∈ Iδ

,

where Πuu(huu(vc,ve)(vuu), vuu) = (vc, ve). Consequently huu(vc,ve)(0) = (vc, ve). Similarly we define the projection

Πss : W ss,eloc (P ) ⊂ R4 → R2 and hss(vc,ve) : R→ R2.

Now we define the addressed transformation T2 as

T2 : (vc, ve, vss, vuu) 7→ (vc, ve, vss, vuu)

:=(

(vc, ve)− huuΠuu(vc,ve,vuu)(vuu)− hssΠss(vc,ve,vss)(v

ss) + Πuu(vc, ve, vuu) + Πss(vc, ve, vss), vss, vuu)

.

Clearly T2 is Ck-smooth. Further let (vc, ve, 0, vuu) ∈ Fuu(vc,ve,0,0), then

T2(vc, ve, 0, vuu) =

(

(vc, ve)− huuΠuu(vc,ve,vuu)(vuu)− hss(vc,ve)(0) + Πuu(vc, ve, vuu) + (vc, ve), 0, vuu

)

=(

(vc, ve)− huu(vc,ve)(vuu) + Πuu(vc, ve, vuu), 0, vuu

)

.

Hence T2(vc, ve, 0, vuu) = (vc, ve, 0, vuu). Similar considerations apply for the leaves Fss(vc,ve,0,0). Hence, after applying

T2, the c- and e-component of any leaf of Fss,Fuu are constant.

Lemma 6.8. The mapping T2 is a transformation.

Proof of Lemma 6.8. We infer from the above considerations that

DT2(vc, ve, 0, 0) =

1 0 ∗ ∗

0 1 ∗ ∗

0 0 1 0

0 0 0 1

,

where the ∗ represent some unspecified terms. Hence the Inverse Function Theorem shows that T2 is a transformation

locally near any point of W eloc(P ). Similarly to the proof of the invertibility of T1 in Lemma 6.2, it follows that T2 is

globally invertible in a neighbourhood of P .

Next we investigate the structure of the transformed vector field f := (f c, fe, fss, fuu).

Lemma 6.9. The Ck-smooth transformation T2 satisfies the following assertions.

(i) The vector field of Lemma 6.2 reads in the by T2 transformed coordinates

f :=

1 + Ec(vc, ve)ve +Ac(vc, ve, vss, vuu)vssvuu

Ae(vc, ve)ve +B(vc, ve, vss, vuu)vssvuu



,

where Ass/uu(vc, 0, 0, 0) = 0 for all vc ∈ S1. Moreover, f depends on µ and λ and is Ck-smooth in v and µ, λ.


(ii) W ssloc(P ) = v

e = 0, vuu = 0 and Wuuloc (P ) = v

e = 0, vss = 0.

(iii) If λ = 0, then the vector field is reversible in the v-coordinates and the reverser R acts as

(vc, ve, vss, vuu) 7→ (−vc, ve, vss, vuu).

Proof of Lemma 6.9. First we verify the form of the vector field f given in (i). We observe that the

coordinates gained by T2 are a special case of the coordinates in Lemma 6.2. Hence the structure of fss and

fuu, which we revealed, is still preserved. Next we consider the e-component, fe, of f . Let v = T2v and let

ϕt = ([ϕt]c, [ϕt]e, [ϕt]ss, [ϕt]uu). Along any fibre Fss(vc,ve,0,0) the values of vc, ve are constant. Together with the flow

invariance of Fss this yields [ϕt]e(vc, ve, vss, 0) = [ϕt]e(vc, ve, 0, 0). Since [ϕt]e(vc, ve, vss, 0) = fe(ϕt(vc, ve, vss, 0)), it

follows that fe(ϕt(vc, ve, vss, 0)) = fe(ϕt(vc, ve, 0, 0)). And in particular for t = 0:

fe(vc, ve, vss, 0) = fe(vc, ve, 0, 0).

Similarly we infer fe(vc, ve, 0, vuu) = fe(vc, ve, 0, 0). With that we find

fe(vc, ve, vss, vuu) = fe(vc, ve, 0, 0) +O(vss vuu) = fe(vc, 0, 0, 0) +O(ve) +O(vss vuu).

Since (vc, 0, 0, 0) ∈ P , it follows that fe(vc, 0, 0, 0) = 0. This gives the structure of fe in (i). Similarly we obtain the

structure of f c.

Next we prove (ii). The strong stable and strong unstable manifolds of P consist of the leaves of the foliations Fss

and Fuu, with base points on P , see Theorem 6.2. Recall that in the v coordinates the leaves of Fss and Fuu are

given by the sets (vc, ve, s, 0) | s ∈ Iδ and (vc, ve, 0, s) | s ∈ Iδ and therefore

W ssloc(P ) = v


e = 0, vss = 0.

It remains to investigate the reversibility of f . In the v-coordinates the reverser acts as

R = T2 R T−12 .

Hence R(T2(vc, ve, vss, vuu)) = T2R(v

c, ve, vss, vuu) = T2(−vc, ve, vuu, vss). We show that

T2(−vc, ve, vuu, vss) = (−vc, ve, vuu, vss).

We introduce the notation

(

Πuu,c(v),Πuu,e(v))

:= Πuu(v),(

Πuu,c(v),Πss,e(v))

:= Πss(v),(

huu,c(vc,ve)(vuu), huu,e(vc,ve)(v

uu))

:= huu(vc,ve)(vuu),

(

hss,c(vc,ve)(vuu), hss,e(vc,ve)(v

uu))

:= hss(vc,ve)(vuu).

According Theorem 6.2 the leaf Fuu(Πuu(v),0,0) is mapped by the reverser R onto the leaf FssR(Πuu(v),0,0). This implies that

R(

ΠuuFuu(Πuu(v),0,0), 0, 0)

= ΠssFssR(Πuu(v),0,0). Consequently, it follows thatR(

Πuu(v), 0, 0)

= Πss(

R(Πuu(v), 0, 0), 0, 0)

.


Hence

−Πuu,c(v) = Πss,c(Rv), Πuu,e(v) = Πss,e(Rv). (6.15)

Moreover, from the uniqueness of huu, hss we infer

huu,cΠuu(v)(vuu) = −hss,c(−Πuu,c(v),Πuu,e(v))(v

uu), huu,eΠuu(v)(vuu) = hss,e(−Πuu,c(v),Πuu,e(v))(v

uu). (6.16)

Exploiting (6.15) and (6.16) we obtain

T2(−vc, ve, vuu, vss)

=(

− vc − huu,cΠuu(−vc,ve,vss)(vss)− hss,cΠss(−vc,ve,vuu)(v

uu) + Πuu,c(−vc, ve, vss) + Πss,c(−vc, ve, vuu),

ve − huu,eΠuu(−vc,ve,vss)(vuu)− hss,eΠss(−vc,ve,vuu)(v

uu) + Πuu,e(−vc, ve, vss) + Πss,e(−vc, ve, vuu), vuu, vss)

=(

− vc + hss,cΠss(vc,ve,vss)(vss) + huu,cΠuu(−vc,ve,vuu)(v

uu)−Πss,c(−vc, ve, vss)−Πuu,c(−vc, ve, vuu),

ve − hss,eΠss(vc,ve,vss)(vss)− huu,eΠuu(−vc,ve,vuu)(v

uu) + Πss,e(−vc, ve, vss) + Πuu,e(−vc, ve, vuu), vuu, vss)

= (−vc, ve, vuu, vss).

This proves the reversibility of the vector field. Finally we rename the variables v back to v, which gives the lemma.

Remark 6.3. The periodic orbit P possess the same Floquet multipliers in the ”new”v-coordinates, as in the cartesian

coordinates.

Consider the representation of the vector field f given in Lemma 6.9. Next we transform the vector field so that

the term Ec(vc, ve) vanishes. As in [65, (3.10.8)], we accomplish this by a rescaling of time. Let b : R4 → R be a

Ck-smooth mapping with b(v) 6= 0, ∀v ∈ R4. Define

τ(t, v) :=

∫ t

0

b(ϕs(v)) ds. (6.17)

For fixed v the function τ(·, v) is strictly monotone, and hence it has an inverse function t(·, v). With that we define

ϕτ (v) := ϕt(τ,v)(v). (6.18)

The function ϕ(·)(v0) solves the initial value problem

v =1

b(v)f(v) =: f(v), v(0) = v0. (6.19)

Using the quantity Ec defined in Lemma 6.9, we determine a specific b(·) by

b(vc, ve, vss, vuu) := 1 + Ec(vc, ve) ve. (6.20)

With this specification the vector field f reads


f :=

1 +Ac(vc, ve, vss, vuu)vssvuu

Ae(vc, ve)ve +B(vc, ve, vss, vuu)vssvuu



,

where we renamed the terms in the transformed vector field back to Ak, k = c, e, ss, uu and B. This representation

can be verified by considering a Taylor expansion of f w.r.t. ve at ve = 0. Note that the choice of b in (6.20) implies

that on the periodic orbit P holds τ(t, P ) = t. Observe that for any (sufficiently small) vicinity U of P there are

positive constants c, C such that c t ≤ τ(t, v(t)) ≤ C t, for any solution v(t) ∈ U of v = f . The vector field f satisfies:

Lemma 6.10. For the vector field f given by (6.19), (6.20) the following assertions are true.

(i) Ass(vc, 0, 0, 0) = Ass(vc, 0, 0, 0) for all vc ∈ S1.

(ii) W ssloc(P ) = v


e = 0, vss = 0.

(iii) For λ = 0, the vector field f is reversible.

(iv) P is still a periodic orbit with period 2π and nontrivial Floquet multipliers e2παss

, e2παe

, e2παuu

.

Proof of Lemma 6.10. First we observe that (i) is a direct consequence of (6.19) and (6.20). By (6.18) assertion

(ii) is immediate. Next we show (iii). According to Lemma 6.9 (iii) we find Ec(vc, ve) = Ec(−vc, ve). This implies

Rf(v) =Rf(v)

1 + Ec(

vc, ve) ve=

−f(Rv)

1 + Ec(

− vc, ve) ve= −f(Rv).

Hence the vector field f is reversible.

Now we turn to (iv). We observe f|P = (1, 0, 0, 0)T , this reveals that P is a 2π-periodic orbit of f . What is

more, the rescaling of time does not change the invariant manifolds of f . We note that f c does not depend on ve.

Therefore, (vc, ve, 0, 0)T | vcfix, ve ∈ R are flow-invariant leaves and (0, 1, 0, 0)T is an eigenvector of Dϕ2π(γ0). From

the definition of ϕ we derive that

Dϕ2π(γ0) = Dϕ2π(γ0) + f(γ0)Dvt(2π, γ0)

= Dϕ2π(γ0)− (1, 0, 0, 0)T(

0,

∫ t(τ,γ0)

0

Ec(

s, 0)Deϕse(γ0) ds, 0, 0

)

. (6.21)

By construction Dϕ2π(γ0) has eigenvectors (1, 0, 0, 0)T , (0, 0, 1, 0)T , (0, 0, 0, 1)T and leaves span(1, 0, 0, 0)T and

span(0, 1, 0, 0)T invariant. Hence Dϕ2π(γ0) has w.r.t. the base (1, 0, 0, 0)T , (0, 1, 0, 0)T , (0, 0, 1, 0)T , (0, 0, 0, 1)T

the matrix representation

1 0 0 0

0 e2παe

0 0

0 0 e2παss

0

0 0 0 e2παuu

.

Together with (6.21) follows (iv) of the lemma.


For our further analysis we rename the rescaled time τ back to t.

Remark 6.4. Recall that Lemma 6.9 yields a transformation (in phase space) that makes the c-component of the

vector field, restricted to W ss,eloc (P ) and W e,uu

loc (P ), independent of vss and vuu. Lemma 6.10 accomplishes the same

effect with respect to the dependence on ve, but uses a rescaling of time. Here we shortly discuss, why in general

there is no transformation in phase space that generates a vector field with a c-component that is independent of ve

in W eloc(P ).

A necessary condition to construct a transformation in phase space that effects the same as the rescaling of time, is

the existence of a smooth, flow invariant foliation of W eloc(P ) with P as base manifold (cf. also the transformation T2

in Lemma 6.9). We see this by inspecting the vector field f in Lemma 6.10 within the manifold W eloc(P ). The sets

vc = const. decompose W eloc(P ) into 1-dimensional manifolds that are mapped by the flow onto one another. Hence

vc = const. defines a smooth, flow invariant foliation of W eloc(P ). But such a foliation does not necessarily exists, if

the stable Floquet multiplier tends to 1.

More precisely we consider the following counterexample, which is inspired by an example of Rellich, [33, Example

5.3]:

Let n ∈ N, on S1 × R we consider the flow

ϕt(vc, ve, λ) =

vc + t−ve sin

(

1(1−λ)

)

2+sin(nvc) ·[

(1− e−1

(1−λ)2 )t/2π − 1]

ve

2+sin(nvc) (1− e−1

(1−λ)2 )t/2π ·(

2 + sin(

nvc + t−ve sin

(

1(1−λ)

)

2+sin(nvc) ·[

(1− e−1

(1−λ)2 )t/2π − 1])

)

that corresponds to the ODE

(

vc

ve

)

=

1−ve sin

(

−1(1−λ)

)

2π (2+sin(nvc)) · ln(1− e−1

(1−λ)2 )

ve

2π · ln(1− e−1

(1−λ)2 ) + ve

2+sin(nvc)

(

1−ve sin

(

−1

(1−λ)2

)

2π(2+sin(nvc)) · ln(1− e−1

(1−λ)2 ))

· cos(nvc)

.

The flow, as well as, the ODE depend smoothly on the parameter λ. Further we see that (vc, 0) | vc ∈ S1 is a periodic

orbit of the ODE with period 2π. The corresponding monodromy matrix Dϕ2π(0, 0, λ) reads

Dϕ2π(0, 0, λ) =

1 e−1

(1−λ)2sin( 1

1−λ)

2+sin(nvc)

0 1− e−1

(1−λ)2

and Dϕ2π(0, 0, 1) =

(

1 0

0 1

)

.

The matrix Dϕ2π(0, 0, λ) is arbitrarily often differentiable with respect to λ. For λ 6= 1 it possesses the eigenvalues

1, 1− e−1

(1−λ)2 . The corresponding eigenvectors are(

1, 0)T

and(

− sin( 11−λ ), 1

)T. Since Dϕ2π(0, 0, λ) is a monodromy

matrix, its eigenvalues correspond to the Floquet multipliers of the periodic orbit. Further 1 − e−1

(1−λ)2 < 1, if λ 6= 0,

hence the periodic orbit possesses a 2-dimensional, stable manifold. This manifold possesses a smooth, flow invariant

foliation, [65]. But these foliations cannot be extended continuously to λ = 1. For this we observe that the eigenvectors(

− sin( 11−λ ), 1

)Tdo not converge for λ→ 1. This contradicts the existence of a smooth, flow invariant foliation of the

stable manifold.

We have seen that in this example a transformation in phase space is not enough to make the c-component of the


vector field independent of ve. Next we investigate the effect of the rescaling of time within the above example. In

Figure 6.6 the periodic orbit (violet) and one solution is drawn. There the parameters are λ = 0.25 and n = 7. The

solution starts at t = 0 in (vc, ve) = (0, 1) and the black dots on the solution curve are drawn after every time step

t = 2π/75. Moreover the red, yellow and orange points give the solution after the time steps t = 2π. Since 2π is the

period of the periodic orbit, these points lie on three single stable fibres.

Figure 6.6: The periodic orbit and the solution starting at the point (vc, ve) = (0, 1) for λ = 0.25, n = 7. The coloureddots lie on the same stable fibres. The graphic is drawn with Maple.

According to (6.17) the transformation of time that makes the c-component the vector field independent of ve reads

τ := t−

∫ t

0

ve sin(

1(1−λ)

)

2π (2 + sin(nvc))· ln(1− e

−1

(1−λ)2 )ds

This equation can be solved with respect to t. The CAS-system Maple yields

t(vc, ve, τ) = τ−ve sin

(

1(1−λ)

)

2 + sin(nvc)

−2π

ln(1− e−1

(1−λ)2 )Lam

(

−ve sin( 1

(1−λ) )

2π(2 + sin(nvc))ln(1− e

−1

(1−λ)2 ) · (1− e−1

(1−λ)2 )τ2π−

ve sin( 1(1−λ)

)

2π(2+sin(nvc))

)

.

Here Lam(·) denotes the upper branch of the Lambert-Function (or Ω-Function). In the rescaled time the c-component

of the vector field is simply 1 for all vc, ve and the solution of the corresponding ODE is given by ϕτ (vc, ve, λ) =

ϕt(vc,ve,τ)(vc, ve, λ). Figure 6.7 shows the solution of Figure 6.6 after the rescaling. The coloured points (that belong

to one fibre) illustrate that the fibres of the periodic orbit has become straight lines, which coincides with the result


that the value of the vc-component of the solution is independent of ve.

Figure 6.7: The solution through (vc, ve) = (0, 1) after rescaling the time. The fibres are now straight lines. Thegraphic is drawn with Maple.

The next transformation involves the first integral H introduced in Hypothesis (H2.8). For λ = 0 the periodic orbit

P lies in the zero level set of H, see Hypothesis (H2.9) (iv). Thus W ssloc(P ) and W

uuloc (P ) lie in the same level set of H:

H(vc, 0, vss, 0, µ) ≡ 0, H(vc, 0, 0, vuu, µ, ) ≡ 0.

Hence

DvcH(vc, 0, 0, 0, µ) ≡ 0 and DvssH(vc, 0, 0, 0, µ) ≡ 0 and DvuuH(vc, 0, 0, 0, µ) ≡ 0.

Exploiting Hypothesis (H2.9) (iv), we get DveH(vc, 0, 0, 0, µ) 6= 0, and due to continuity

DveH(vc, ve, vss, vuu, µ) 6= 0,

for sufficiently small ve, vss, vuu. This allows to introduce new coordinates

(

vc, ve, vss, vuu)

:=(

vc, H(vc, ve, vss, vuu, µ), vss, vuu)

=: TH(vc, ve, vss, vuu, µ). (6.22)


Since H is R invariant, see (H2.8), the reverser R acts on the new coordinates

R(

vc, ve, vss, vuu)

=(

− vc, ve, vuu, vss)

.

Lemma 6.11. The Ck-transformation defined in (6.22) transforms the vector field of Lemma 6.10 in

f :=

1 +Ac(vc, ve, vss, vuu) vssvuu

λ[

Ae(vc, ve) ve +B(vc, ve, vss, vuu) vssvuu]

(

αss +Ass(vc, ve, vss, vuu))

vss(

αuu +Auu(vc, ve, vss, vuu))

vuu

.

Furthermore Ass(vc, 0, 0, 0) = Auu(vc, 0, 0, 0) = 0 for all vc ∈ S1.

Proof of Lemma 6.11. First we note that due to the structure of the transformation TH the e-, ss- and uu-

component of f remains unchanged if DTH acts on f . Next we consider the e-component, fe, of f . Let λ = 0 and let

ϕt(v) be an orbit of the vector field f . Then, due to (6.22),

(

ϕt(v))e≡ const.

Hence, for λ = 0, we find that fe(v) = 0. Renaming v into v gives the Lemma.

Remark 6.5. Due to the R-invariance of H the vector field f is R-reversible.

A final transformation takes Ae(vc, ve), cf. the representation of the vector field f in Lemma 6.11, in the form

Ae(vc, ve) = αe +Ae(vc, ve) ve.

To this end we modify the ve-coordinate. Basically we proceed as in the definition of ess/uuγt , cf. (6.1). The introduction

of these quantities permits to isolate the leading order terms αss, αuu in the vss- and vuu-coordinates. We define

(

vc, ve, vss, vuu)

= T3(

vc, ve, vss, vuu)

:=(

vc,eα

evc

Deϕvc

e (0)ve, vss, vuu

)

. (6.23)

Lemma 6.12. The transformation defined in (6.23) transforms f into

f :=

1 +Ac(vc, ve, vss, vuu) vssvuu

λ[

(αe +Ae(vc, ve) ve) ve +B(vc, ve, vss, vuu) vssvuu]

(

αss +Ass(vc, ve, vss, vuu))

vss(

αuu +Auu(vc, ve, vss, vuu))

vuu

.

Proof of Lemma 6.12. We observe that the structure of the transformation of T3 implies that the e-, ss- and

uu-component of f remains unchanged if DTH acts on f . In what follows we restrict our considerations to the manifold


W e(P ), then it holds that

f(vc, ve, 0, 0) = DT3(vc, ve, 0, 0) f(vc, ve, 0, 0).

The derivative of T3 reads

DT3(v) =

1 0 0 0

a eαe(vc−2π) 0 0

0 0 1 0

0 0 0 1

,

with a = αe eαevc

Deϕvce (0)

ve − eαevc

[Deϕvce (0)]2

Deϕvc

e (0) ve. Combining the above lines we get

f(vc, ve, 0, 0) = DT3(vc, ve, 0, 0)

(

1, λ Ae(vc, ve), 0, 0)

=(

1, αeve −eα

evc

[Deϕvc

e (0)]2Deϕ

vc

e (0) ve + λ Ae(vc, ve) ve, 0, 0)

.

We observe that Deϕvc

e (0) = λ Ae(vc, 0)Deϕvc

e (0). This and a Taylor expansion yields:

fe(vc, ve, 0, 0) = αeve −eα

evc

[Deϕvc

e (0)]λ Ae(vc, 0) ve + λ Ae(vc, ve) ve

= αeve + λ Ae(vc, ve) ve · ve

= αeve + λ Ae(

vc, e−αevc Deϕ

vc

e (0) ve)

e−αevc Deϕ

vc

e (0) ve · ve

= αeve + λAe(vc, ve) (ve)2,

for some smooth function Ae. Finally we rename the variables v.

Remark 6.6. The vector field f is R- reversible and the reverser R acts on (vc, ve, vss, vuu) as

R(vc, ve, vss, vuu) = (−vc, ve, vuu, vss).

This finishes the proof of Theorem 2.3.

99

Chapter 7

The Shilnikov problem

This chapter is devoted to the proof of Theorem 2.4. We prove the existence of a solution of the Shilnikov problem

near the periodic orbit P . Further we derive the leading terms of the single components of this solution. As claimed

in the Theorem 2.4, it turns out that the leading order terms are determined by the Floquet exponents of P . Here we

consider the case where P possesses Floquet exponents

αss(µ, λ) < αe(µ, λ) ≤ 0 < αuu(µ, λ).

The proof applies the ideas of Lin’s Method [62]. More precisely we exploit the analysis of Krupa, Sandstede and

Szmolyan [44, Section 5, Theorem 4]. There a Shilnikov problem near a slow manifold of a singular perturbed system

has been considered. The slow manifold possesses 1-dimensional stable and unstable manifolds and a centre manifold

of arbitrary dimension. The differences to our present situation are

• The vc-component of the solution of the Shilnikov problem is prescribed at t = 0 “in the middle” of the time

interval [−L,L]. (In contrast to [44], where the vc-component is determined at t = −L).

• vs := (ve, vss) is a 2-dimensional stable component (instead of an 1-dimensional component as in [44]), and what

is more we are in need of the leading order terms of both, the weak and the strong stable coordinate ve(L) and

vss(L) of v(L).

• The weak stable Floquet exponent αe = λαe tends to zero as λ→ 0.

According to the last point, the dimension of the involved manifolds differ at λ 6= 0 and λ = 0. This distinguishes our

problem here qualitatively from the considerations in [44]. Nevertheless we use methods of [44] extensively.

Let us recall some definitions from Section 2.3. We denote by ν > 0 some arbitrarily small constant and define



|αe(µ, λ)| .

We consider a system of differential equations that possesses the structure of (2.8),

vc = 1 +Ac(v, λ)vssvuu

ve = λ [(αe(λ) + Ae(vc, ve, λ) ve) ve +B(v, λ) vss vuu]

vss = (αss(λ) +Ass(v, λ)) vss

vuu = (αuu(λ) +Auu(v, λ)) vuu

=: f(v, λ),

100 7 The Shilnikov problem

where v = (vc, ve, vss, vuu) ∈ S1 × R3. Observe that for the reason of brevity we suppress the µ dependency in the

notation. We assume

Ass(vc, 0, 0, 0, λ) = Auu(vc, 0, 0, 0, λ) ≡ 0, ∀ vc ∈ S1.

For the reader’s convenience we repeat Theorem 2.4:

Theorem (Theorem 2.4). Assume Hypothesis (H2.7)-(H2.9). There exists positive constants L0, δ0, λ0 such that for

any L > L0 > 0, ϕ ∈ S1, δ0 > δ > 0, λ ∈ Jλ, |λ| < λ0 and all µ ∈ Jµ the boundary value problem

v = f(v, µ, λ), where vss(−L) = δ, vc(0) = ϕ, ve(−L) = λ ξe, vuu(L) = δ

has a unique smooth solution v(·) = v(·, µ, λ) = (vc(·, µ, λ), ve(·, µ, λ), vss(·, µ, λ), vuu(·, µ, λ)) on [−L,L]. Moreover,

the following estimates hold:

v(−L) =(

ϕ− L, λ ξe, δ, ∆uue−2αuuL)

+(

O(

e−(αuu+α)L)

, 0, 0, λO(

e−(2αuu−αe)L)

+O(

e−(2αuu+α)L)

)

,

v(L) =(

L+ ϕ, λ ξe∆ee2αeL, ∆sse2α

ssL, δ)

+(

O(

e(αss−α)L

)

, λ2O(

e(αe+2αe)L

)

+ λO(

e−L)

, λO(

e(2αss+αe)L

)

+O(

e(2αss−α)L

)

0)

,

where ∆ss = ∆ss(L, δ, ϕ, ξe, µ, λ), ∆e = ∆e(L,ϕ, ξe, µ, λ), ∆uu = ∆uu(L, δ, ϕ, ξe, µ, λ) are some smooth, positive,

bounded functions. Moreover ∆ss = ∆ss0 (ϕ, ξe, δ, µ, λ) + λ

[

∆sse (ϕ, ξe, µ, λ) + O

(

eαeL)]

, for some smooth, bounded

non-zero functions ∆ss0 ,∆

sse . Furthermore, if λ = 0, then ∆uu is independent of L and ∆e ≡ 1. The solution v(·) is

smooth in z = L,ϕ, ξe, µ and the following estimates hold true

Dz v(−L, ) =( d

dz(ϕ− L) +O

(

e−2αL)

,d

dz(λ ξe), 0, O

(

e−2αuuL)

)

,

Dz v(L) =( d

dz(ϕ+ L) +O

(

e−2αL)

, λO(

e2αeL)

, O(

e2αssL)

, 0)

.

If z = λ, then the same estimates as above hold true, for vc, vss, vuu. The ve component satisfies the following

estimates. If λ 6= 0, then Dλ ve(L) = d

dλ (λ ξe∆ee2α

eL) + λO(

e3αeL)

+ O(

e−L)

and if λ = 0, then the (one-side)

derivative Dλ ve(L) = ξe +O

(

e−αL)

.

Moreover, estimates for higher derivatives of v(·) can be obtained by differentiating the estimates of the first derivatives.

Remark 7.1. The estimate of the derivatives of the vss and vuu components are rather rough, but sufficient for the

”Snaking analysis” in the previous chapters. In fact, the proof given here would yield even better estimates for the

derivatives of vss and vuu.

Theorem 2.4 is the counterpart to [44, Theorem 4] or [3, Lemma 3.1], respectively. In [44], Krupa et al. consider

the Shilnikov problem in the context of slow/fast systems, where the slow manifold possesses a normally hyperbolic

structure. Furthermore, Krupa et al. give exact estimates of the derivatives of all components of v.

7.1 Outline of the proof 101

7.1 Outline of the proof

In this first section we give a rough outline of the proof. To enhance the readability we suppress the dependence on µ

and frequently the λ dependency in the notation.

Let v be the solution according to Theorem 2.4. In particular v connects the cross-sections Σin and Σout with a flight

time 2L. We conceive v as being composed of two functions v+ and v−:

v(t) :=

v+(t+ L), t ∈ [−L, 0],

v−(t− L), t ∈ [0, L],v+(L) = v−(−L).

According to the boundary conditions we impose on v, the solutions v+ and v− satisfy

vc+(L) = ϕ, ve+(0) = λξe, vss+ (0) = δ, vc−(−L) = ϕ, vuu− (0) = δ. (7.1)

We introduce I+ := [0, L] and I− := [−L, 0] write

v+(t) =: q+(t) +X+(t), t ∈ I+ and v−(t) =: q−(t) +X−(t), t ∈ I−, (7.2)

where q+ and q− are located within the (extended) stable manifold W ss,eloc (P ) and the (strong) unstable manifold

Wuuloc (P ). In particular this means that

quu+ (t) ≡ 0 and qe−(t) ≡ 0, qss− (t) ≡ 0,

respectively. Furthermore we claim that q+ and q− satisfy the additional boundary conditions

qss+ (0) = δ, qe+(0) = λ ξe, qc+(L) = ϕ and qc−(−L) = ϕ, quu− (0) = δ. (7.3)

In Section 7.2 we give estimates of the reference solutions q±. According to (7.1) and (7.2) the boundary conditions (7.3)

together with the coupling condition v+(L) = v−(−L) translate into the following boundary and coupling conditions

for the functions X±:

(i) Xc+(L) = 0, (ii) Xe

+(0) = 0, (iii) Xss+ (0) = 0, (iv) Xuu

− (0) = 0. (7.4)

X+(L)−X−(−L) =(

0, −qe+(L), −qss+ (L), quu− (−L)

)

. (7.5)

We show the existence of X± via the Banach Fixed Point Theorem. In that respect the main technical obstacle is to

derive the corresponding operator equation. To obtain this equation we plug v±(t) = q±(t) +X±(t) into (2.8) and get

v±(t) = f(v±(t)) = f(

q±(t) +X±(t))

,

where f was defined as the right hand side of (2.8). On the other hand v±(t) = q±(t) + X±(t) = f(q±(t)) + X±(t).

Combining both expressions yields that X±(t) has to satisfy the equation:

X±(t) = f(

q±(t) +X±(t))

− f(q±(t)) = Df(q±(t))X±(t) + h(t,X±), (7.6)


for some appropriate function h that is w.r.t. X± of order O(

(X±)2)

. Moreover, the structure of (2.8) implies that

the e-component of h vanishes for λ = 0. To solve (7.6) we investigate at first the linearised inhomogeneous equation

Y±(t) = Df(

q±(t))

· Y±(t) + g±(t), (7.7)

where g±(t) are taken out of certain functions spaces V±, of continuous functions “satisfying” exponential rates corre-

sponding to the exponents αe, αss and αuu. In Section 7.3 we study the corresponding homogeneous equation to (7.7).

Then, in Section 7.4 we show that for any g± there is a solution Y±, which meets the boundary conditions (7.4). More

precisely we show that:

For any g± ∈ V± and any d = (0, de, dss, duu) ∈ R4, there is a solution Y± of (7.7) that satisfies the boundary conditions

(i) Y c+(L) = 0, (ii) Y e+(0) = 0, (iii) Y ss+ (0) = 0, (iv) Y uu− (0) = 0, (7.8)

as well as, the coupling condition

Y+(L)− Y−(−L) = (0, de, dss, duu). (7.9)

With that we define for fixed L > L0 the operator NL that maps onto the solutions Y±:

NL(d, g) := (Y+, Y−), g := (g+, g−)

and the Nemitskii operators, see Section 7.5,

G±(X±)(t) := f(

q±(t) +X±(t))

− f(

q±(t))

−Df(

q±(t))

·X±(t), t ∈ I±, X± ∈ V±.

The concatenated operator NL G, where G(X) := (G+(X+), G−(X−)), maps the space V into itself. We consider

the fixed point equation

X± = NL(d,G(X)).

By choosing d = (0,−qe+(L),−qss+ (L), quu− (−L)) any solution of this equation solves (7.6) with the boundary conditions

(7.4) and the coupling condition (7.5). The Banach Fixed Point Theorem yields for any sufficiently small (d, ϕ, ξe, δ, λ)

a unique fixed point X of this equation. By the Implicit Function Theorem this fixed point depends smoothly on

(d, ϕ, ξe, δ, µ, λ). Furthermore we show that X depends also smoothly on L. Recalling (7.2), this proves the existence

of a solution of the Shilnikov problem, as it is claimed in Theorem 2.4. It remains to derive the estimates of this

solution. To this end one has to estimate X and in particular X(0). This is done in Section 7.6 by exploiting the

structure of Y± and V±.

7.2 The reference solutions q±

As described in the previous outline of the proof, we compose v± = q±+X±, where q± denote solutions of (2.8) within

the (extended) stable manifold W ss,eloc (P ) and the unstable manifold Wuu

loc (P ), respectively. In this section we derive

these solutions q±.

7.2 The reference solutions q± 103

Technical results

The following lemma and its proof contain typical arguments we will use frequently in the course of our analysis. The

proof of this lemma is similar to the proofs of [62, Lemma 1.5] or [44, Lemma 1], respectively.

Lemma 7.1. Consider the scalar differential equation

x = αx+B(t)x, (7.10)

with α < 0, and assume that there are constants β < 0 and C such that |B(t)| < Ceβt. Then, for any solution x(·) of

(7.10) the following is true:

• There exits an η such that limt→∞ e−αtx(t) = η.

• There exists a constant C such that |x(t)− eαtη| ≤ Ce(β+α)t.

Proof of Lemma 7.1. Using the variation of constants formula we rewrite (7.10) as into the integral equation

x(t) = eα(t−s)x(s) +

∫ t

s

eα(t−τ)B(τ)x(τ) dτ. (7.11)

First we show that the limit limt→∞ e−αtx(t) does exist: To this end we first note that for each α > α there is a

constant D such that |x(t)| ≤ Deαt. The existence of such a α follows from the theory of exponential dichotomies [13].

Now, multiplying (7.11) by e−αt yields

|e−αtx(t)− e−αsx(s)| ≤

∫ t

s

e−ατ |B(τ)||x(τ)| dτ ≤ CD

∫ t

s

e(β+α−α)τ dτ.

Here we assume that s < t. If α is close enough to α then β + α−α < 0, and thus |e−αtx(t)− e−αsx(s)| can be made

arbitrarily small by choosing s and t large enough.

Next we determine x(t)− eαtη by considering the limit s→∞ in (7.11)

x(t) = eαtη +

∫ t

∞

eα(t−τ)B(τ)x(τ) dτ.

Since the limit limt→∞ e−αtx(t) does exist, there is a constant D such that |x(t)| ≤ Deαt. With that we find

|x(t)− eαtη| ≤ CDeαt∫ t

∞

eβτ dτ = Ce(β+α)t.

Corollary 7.2. Consider the scalar differential equation (7.10) with α > 0 and β < 0, and denote the corresponding

transition matrix by Φ(·, ·). Then, for each w ∈ R there exist an η, and there exists a constant C such that for all

L > 0

|Φ(0, L)w − e−αLη| ≤ Ce(β−α)L.

Proof of Corollary 7.2. We first note that for each α with 0 < α < α there is a D such that Φ(0, L)w ≤ De−αL.

(In a somewhat more general context this follows from a roughness theorem for exponential dichotomies.) Denote the


right-hand side of (7.10) by A(t). In these terms, Φ(t, 0)Φ(0, t) = 1 yields Φ(0, t) = −Φ(0, t)A(t) = −A(t)Φ(0, t); the

latter equality is due to the fact that (7.10) is a scalar equation. Now, the statement follows with Lemma 7.1.

Corollary 7.3. Consider the scalar differential equation

x = αx+B(t)x+ E(t),

with α < 0, and assume that there are constants β < 0 and C such that |B(t)| < Ceβt and |E(t)| ≤ Ceαt. Let t0 ≥ 0

be fixed and x0 ∈ R. Then, the solution x(·) of the initial value problem (7.10) with x(t0) = x0 satisfies

x(t) = O(

eαt)

, ∀t ≥ t0, ∀α < α.

Proof of Corollary 7.3. Denote by Φ(·, ·) the transition matrix of the homogeneous equation x = αx+B(t)x. By

the variation of constant formula

x(t) = Φ(t, t0)x0 +

∫ t

t0

Φ(t, τ)E(τ) dτ.

By Lemma 7.1 it follows that |Φ(t, t0)x0| = O(eαt). The claim follows by exploiting the estimate of |C(τ)|.

The solutions q±

Now we stipulate the solutions q±. Restricted to W ss,eloc (P ) = vuu = 0, Equation (2.8) reads

vc = 1

ve = λ(

αe +Ae(vc, ve) ve)

ve

vss =(

αss +Ass(vc, ve, vss, 0))

vss (7.12)

vuu = 0.

Restricting (2.8) to the manifold Wuuloc (P ) = v

e = 0, vss = 0 we obtain

vc = 1

ve = 0

vss = 0 (7.13)

vuu =(

αuu +Auu(vc, 0, 0, vuu))

vuu.

We search for solutions q+ and q− of (7.12) and (7.13), respectively, satisfying the following boundary conditions.

qc+(L) = ϕ,

qe+(0) = λ ξe,

qss+ (0) = δ,

quu+ (0) = 0,

qc−(−L) = ϕ,

qe−(−L) = qe+(L),

qss− (0) = 0,

quu− (0) = δ.

(7.14)

In particular q+(t), q−(t− 2L) satisfy the boundary conditions (7.3).


Lemma 7.4. On R+0 and R−

0 there exist unique, smooth solutions q+ of (7.12) and q− of (7.13), satisfying the

boundary conditions (7.14). Moreover, there are smooth, bounded, nonzero functions ∆e = ∆e(L,ϕ, ξe, µ, λ), ∆ss =

∆ss(L,ϕ, ξe, δ, µ, λ), ∆uu = ∆uu(L,ϕ, ξe, δ, µ, λ) such that:

qc+(t) = t− L+ ϕ

qe+(t) = λ ξe∆eeαet + λ2O

(

e2αet)

qss+ (t) = ∆sseαsst +O

(

e2αsst)

quu+ (t) = 0

qc−(t) = t+ L+ ϕ

qe−(t) = 0

qss− (t) = 0

quu− (t) = ∆uueαuut +O

(

e2αuut)

Furthermore ∆ss = ∆ss0 (L,ϕ, ξe, δ, µ, λ) + λ

[

∆sse (L,ϕ, ξe, δ, µ, λ) +O

(

eαet)]

, where both ∆ss0 , ∆

sse are smooth bounded

functions. If λ = 0, then ∆e ≡ 1. For z = L,ϕ, ξe, µ, λ it holds

Dz qss+ (t) = O

(

eαsst)

, Dz quu− (t) = O

(

eαuut)

,

and for z = L,ϕ, ξe, µ

Dz qe+(t) =

d

d z(λ ξe∆eeα

et) + λ2O(

e2αet)

Dλ qe+(t) =

d

d λ(λ ξe∆eeα

et) + λO(

e2αet)

Further , if λ = 0, then Dλ qe+(t) = ξe. Moreover, estimates of higher derivatives of q± can be obtained by differentiating

the estimates of the first derivatives.

Observe that due to the boundary conditions (and in contrast to [44]) qc± depend on L.

Proof of Lemma 7.4. For q− the proof proceeds exactly as in [44], therefore we merely derive the estimates for

q+. At first we observe that qc+(t) = t + ϕ − L is obviously the solution to qc+ = 1, qc+(L) = ϕ. Next we consider the

e-equation:

qe+ = λ(

αe +Ae(t+ ϕ− L, qe+) qe+

)

qe+, qe+(0) = λξe. (7.15)

Let λ < 0. By choosing |ξe| sufficiently small, we infer that |qe+(t)| ≤ Ceνt for some ν < 0 and C > 0. Hence

Lemma 7.1 yields that there is a constant C > 0 such that

|qe+(t, t0, λξe)| ≤ |λ ξe| Ceλα

e(t−t0), (7.16)

where ξe is chosen close enough to zero. Observe that the factor |λ ξe| in (7.16) is due to that fact that qe+ ≡ 0, if

λξe = 0. The variation of constant formula yields for t0 ∈ [0,∞)

qe+(t, t0, qe+(t0, 0, λξ

e)) = eαe(t−t0)qe+(t0, 0, λξ

e) + λ

∫ t

t0

eαe(t−σ)Ae(σ + ϕ− L, qe+(σ, 0, λξ

e)) (qe+(σ, 0, λξe))2dσ.

Exploiting (7.16), we find that there is a positive constant C such that for all t ∈ [0,∞) holds

∣

∣

∣λ

∫ t

t0


e)) (qe+(σ, 0, λξe))2dσ

∣

∣

∣ ≤ C |λ3 (ξe)2|

∫ t

t0

eαe(t+σ)dσ

=λ2 (ξe)2 C

αe∣

∣e2αet − eα

e(t+t0)∣

∣. (7.17)


Next we define

∆e := limt0→∞

1

λ ξee−α

et0 · qe+(t0, 0, λξe).

To see that this limit exists, we consider the variation of constant formula for the initial value qe+(0, 0, λξe) = λξe:

λξe = e−αet0qe+(t0, 0, λξ

e) + λ

∫ 0

t0

−eαeσAe(σ + ϕ− L, qe+(σ, 0, λξ

e)) (qe+(σ, 0, λξe))2dσ, t0 ∈ [0,∞).

Thus by (7.17) we get

∆e = limt0→∞

(

1−1

ξe

∫ 0

t0

−eαeσAe(σ + ϕ− L, qe+(σ, 0, λξ

e)) (qe+(σ, 0, λξe))2dσ

)

≤ 1−λ ξe C

αe.

Therefore ∆e is well defined. Moreover, if λ = 0, then the integral term in the above representation vanishes, see

(7.16). Consequently ∆e ≡ 1. Finally we consider the limit t0 →∞ in the variation of constant formula, which gives

qe+(t, 0, λξe) = λ ξe∆eeα

et + λ

∫ t

∞


e)) (qe+(σ, 0, λξe))2dσ. (7.18)

Exploiting again (7.17) yields the estimate of qe+. Next we estimate qss+ , which solves

qss+ =[

αss +Ass(t+ ϕ− L, qe+, qss+ , 0)

]

qss+ , qe+(0) = λξe, qss+ (0) = δ.

As for qe+, we obtain by Lemma 7.1 that there is a positive constants C such that

|qss+ (t, t0, δ)| ≤ C δ eαss(t−t0). (7.19)

The variation of constant formula reads for qss+ :

qss+ (t, 0, δ) = eαss(t−t0)qss+ (t0, 0, δ) +

∫ t

t0

eαss(t−σ)Ass(σ + ϕ− L, qe+(σ, 0, λξ

e), qss+ (σ, 0, δ), 0) qss+ (σ, 0, δ) dσ

= eαss(t−t0)qss+ (t0, 0, δ) +

∫ 0

t0


e), 0) qss+ (σ, 0, δ) dσ (7.20)

+

∫ t

0


e), 0) qss+ (σ, 0, δ) dσ

+

∫ t

t0

eαss(t−σ)O

(

(qss+ (σ, 0, δ))2)

dσ, ∀t0 ∈ [0,∞).

Similar to (7.17) we obtain that there is a C > 0 such that:

∣

∣

∣

∫ t

t0

eαss(t−σ)O

(

(qss+ (σ, 0, δ))2)

dσ∣

∣

∣ ≤C

|αss|

∣

∣e2αsst − eα

ss(t−t0)∣

∣.


Since Ass(σ+ϕ−L, 0, 0) ≡ 0 and qe+(σ, 0, λξe) = λO

(

eαeσ)

, a Taylor expansion shows that there is a C > 0 such that

∣

∣

∣

∫ t

t0


e), 0) qss+ (σ, 0, δ) dσ∣

∣

∣ ≤ C · eαsst∣

∣

∣λ

∫ t

t0

eαeσ dσ

∣

∣

∣ =C eα

sst

αe∣

∣eαet − eα

et0∣

∣. (7.21)

Next we define

∆ss0 := lim

t0→∞

(

e−αsst0qss+ (t0, 0, δ) +

∫ 0

t0

e−αssσAss(σ + ϕ− L, qe+(σ, 0, λξ

e), 0) qss+ (σ, 0, δ) dσ)

.

As before we exploit the variation of constant formula, (7.20), at t = 0 and that qss+ (0, 0, δ) = δ to obtain

∆ss0 = lim

t0→∞

(

δ −

∫ 0

t0

e−αssσO

(

(qss+ (σ, 0, δ))2)

dσ)

≤ limt0→∞

( C

|αss|

∣

∣1− eαsst0∣

∣

)

=C

|αss|.

Thus ∆ss0 exists and by (7.20) together with (7.21) we find:

qss+ (t, 0, δ) = eαsst∆ss

0 +

∫ t

0


e), 0) qss+ (σ, 0, δ) dσ

+

∫ t

∞

eαss(t−σ)O

(

(qss+ (σ, 0, δ))2)

dσ.

Proceeding as above for qe+ we derive the estimates of qss+ for any fixed λ. Further we investigate the limit of qss+ for

λ→ 0. Note that it is immediate that ∆ss0 is uniformly bounded also for λ→ 0. The same applies to the integral term

∫ t

∞ eαss(t−σ)O

(

(qss+ (σ, 0, δ))2)

dσ in the above representation of qss+ . By exploiting the estimate (7.21), we see that the

remaining integral in this representation tends to zero, if λ→ 0. Therefore we find

qss+ (t, 0, δ) =[

∆ss0 + λ

[

∆sse (L,ϕ, ξe, δ, λ) +O

(

eαet)]]

eαsst +O

(

e2αsst)

= ∆ss eαsst +O

(

e2αsst)

,

where ∆sse (L,ϕ, ξe, δ, λ) is a smooth, bounded function.

Finally we consider exemplarily the derivative of qe+ with respect to λ. The right hand side of (7.15), as well as, the

initial value of qe+(0) depend smoothly on z = L,ϕ, ξe, µ, λ. Thus it is clear that qe+ is differentiable with respect to z.

To obtain the estimates of the derivatives we differentiate (7.15) with respect to λ:

d

dt

(

Dλqe+

)

= λ[

αe + 2Ae(t+ ϕ− L, qe+) qe+ +DeA

e(t+ ϕ− L, qe+) (qe+)

2]

·Dλqe+

+[

αe +Ae(t+ ϕ− L, qe+) qe+ + λDλA

e(t+ ϕ− L, qe+) qe+

]

· qe+. (7.22)

Since |qe+(t)| ≤ λC eαet for some C > 0, Equation (7.22) meets the prerequisites of Corollary 7.3. This proves that

if λ < 0, then Dλqe+(t) = O

(

eαet)

with |αe| < |αe|. Now we differentiate (7.18) and estimate the result of that by

exploiting the estimates on qe+, (7.16), and the above derived estimate of Dλqe+ to obtain the statement of the lemma

for λ < 0.

Finally we observe that for all λ ≤ 0 the initial value qe+(0) ≡ λξe and hence Dλq

e+(0) = ξe. Now let λ = 0: Then the

right hand side of (7.22) is zero. Consequently Dλqe+ is constant. Since Dλq

e+(0) ≡ ξ

e, it follows that Dλqe+(t) ≡ ξ

e.

Similarly we derive the remaining derivatives. This finishes the proof of the lemma.


7.3 The linearised equation along q±

In this section we derive the solutions of the variational equations along q±:

X± = Df(q±)X±.

In the following we show that these variational equations possess exponential trichotomies with corresponding projec-

tions and derive estimates of these projections. The calculations follow closely the lines of [44].

Variational equations along q±

The variational equation of (2.8) along the solution q+(t), t ≥ 0 reads:

Xc+ = Ac(q+)q

ss+X

uu+ (7.23c)

Xe+ = λ

[

αe + 2Ae(qc+, qe+)q

e+ +DeA

e(qc+, qe+)(q

e+)

2]

Xe+ + λDcA

e(qc+, qe+)(q

e+)

2Xc+ + λB(q+) q

ss+ Xuu

+ (7.23e)

Xss+ =

[

αss +Ass(q+) +DssAss(q+) q

ss+

]

Xss+

+DcAss(q+) q

ss+ Xc

+ +DeAss(q+) q

ss+ Xe

+ +DuuAss(q+) q

ss+ Xuu

+ (7.23s)

Xuu+ = [αuu +Auu(q+)]X

uu+ . (7.23u)

Let Φ+(·, ·) be the transition matrix of (7.23). Further let Φe+(·, ·), Φss+ (·, ·), Φuu+ (·, ·) be the transition matrices of the

truncated equations.

˙Xe+ = λ

[

αe + 2Ae(qc+, qe+)q

e+ +DeA

e(qc+, qe+)(q

e+)

2]

Xe+ (7.24e)

˙Xss+ = [αss +Ass(q+)] X

ss+ +DssA

ss(q+) qss+ Xss

+ (7.24s)

˙Xuu+ = [αuu +Auu(q+)] X

uu+ . (7.24u)

Further we denote by X+(·, τ, ζ) = (Xc+(·, τ, ζ), X

e+(·, τ, ζ), X

ss+ (·, τ, ζ), Xuu

+ (·, τ, ζ)) a solution of (7.23), which satisfies

the initial condition X+(τ, τ, ζ) = ζ = (ζc, ζe, ζss, ζuu). We see that (7.23u) coincides with (7.24u). Hence the solution

of (7.23u) is simply given by

Xuu+ (t, τ, ζ) = Φuu+ (t, τ)ζuu. (7.25)

Plugging (7.25) into (7.23c) allows us to integrate the Xc-equation. We obtain

Xc+(t, τ, ζ) = ζc +

∫ t

τ

Ac(q+(σ))qss+ (σ)Xuu

+ (σ, τ, ζ)d σ. (7.26)

Next we consider the Xe-equation (7.23e). Exploiting the representation of the solution of the truncated equation

(7.24e) the variation of constant formula yields

Xe+(t, τ, ζ) = Φe+(t, τ)ζ

e + λ

∫ t

τ

Φe+(t, σ)[

DcAe(qc+(σ), q

e+(σ))(q

e+(σ))

2Xc+(σ, τ, ζ)

+B(q+(σ)) qss+ (σ)Xuu

+ (σ, τ, ζ)]

d σ. (7.27)

7.3 The linearised equation along q± 109

In the same way we obtain the solution of (7.23s).

Xss+ (t, τ, ζ) = Φss+ (t, τ)ζss +

∫ t

τ

Φss+ (t, σ)(

DuuAss(q+(σ)) q

ss+ (σ)Xuu

+ (σ, τ, ζ)

+DeAss(q+(σ)) q

ss+ (σ)Xe

+(σ, τ, ζ) +DcAss(q+(σ)) q

ss+ (σ)Xc

+(σ, τ, ζ))

dσ. (7.28)

The variational equation along q−(t), t ≤ 0 is given by

Xc− = Ac(q−)q

uu− Xss

−

Xe− = λ

[

αeXe− +B(q−) q

ss− Xuu

−

]

Xss− = [αss +Ass(q−)]X

ss− (7.29)

Xuu− =

[

αuu +Auu(q−) +DuuAuu(q−) q

uu−

]

Xuu−

+DcAuu(q−) q

uu− Xc

− +DeAuu(q−) q

uu− Xe

− +DssAuu(q−) q

uu− Xss

− .

We define Φ−(·, ·) as the transition matrix of that variational equation (7.29). And in the same way as above we find

that the solution X−(t, τ, ζ) = (Xc−(t, τ, ζ), X

e−(t, τ, ζ), X

ss− (t, τ, ζ), Xuu

− (t, τ, ζ)) of (7.29) is given by

Xss− (t, τ, ζ) = Φss− (t, τ)ζss,

Xc−(t, τ, ζ) = ζc +

∫ t

τ

Ac(q−(σ))quu− (σ)Xss

− (σ, τ, ζ)d σ,

Xe−(t, τ, ζ) = eα

e(t−τ)ζe + λ

∫ t

τ

eαe(t−σ)B(q−(σ)) q

uu− (σ)Xss

− (σ, τ, ζ)d σ, (7.30)

Xuu− (t, τ, ζ) = Φuu− (t, τ)ζuu +

∫ t

τ

Φuu− (t, σ)(

DssAuu(q−(σ)) q

uu− (σ)Xss

− (σ, τ, ζ)

+DeAuu(q−(σ)) q

uu− (σ)Xe

−(σ, τ, ζ) +DcAuu(q−(σ)) q

uu− (σ)Xc

−(σ, τ, ζ))

dσ,

where Φss− (·, ·), Φuu− (·, ·) denote the transition matrices of

˙Xss− = [αss +Ass(q−)] X

ss− and ˙Xuu

− = [αuu +Auu(q−)] Xuu− +DuuA

uu(q−) quu− Xuu

− .

Projections related to exponential trichotomies of the variational equations

Equations (7.23) and (7.29) are variational equations along the solutions q+, q− that lie within the (extended) sta-

ble and unstable manifolds of the periodic orbit. According to [2], these equations possess exponential trichotomies,

P c±, Pe± + P ss± , Puu± , on R+ and R−, respectively. The corresponding projections are determined by:

P c+(0)ζ = (ζc, 0, 0, 0), P c−(0)ζ = (ζc, 0, 0, 0),

P e+(0)ζ = (0, ζe, 0, 0), P e−(0)ζ = (0, ζe, 0, 0)

P ss+ (0)ζ = (0, 0, ζss, 0), P ss− (0)ζ = (0, 0, ζss, 0),

Puu+ (0)ζ = (0, 0, 0, ζuu), Puu− (0)ζ = (0, 0, 0, ζuu), (7.31)


where ζ = (ζc, ζe, ζss, ζuu) ∈ R4 and

P k±(t) := Φ±(t, 0)Pk±(0)Φ±(0, t), k = c, e, ss, uu.

Consequently the projections commute with the transition matrices Φ±.

Lemma 7.5. Let t ≥ 0. The projections P k+(t), k = c, e, ss, uu satisfy the following estimates:

P c+(t)ζ =(

ζc +O(

e−αt)

ζuu, λ2O(

eαet (|ζc|+ e−αt|ζuu|)

)

,O(

eαsst (|ζc|+ |ζuu|)

)

, 0)

,

P e+(t)ζ =(

0, ζe + λ2 O(

eαet)

ζc + λO(

e−αt)

ζuu,O(

eαsst(λ2 |ζc|+ e−α

et |ζe|+ |λ| e−t |ζuu|))

, 0)

,

P ss+ (t)ζ =(

0, 0, ζss +O(

eαsst (|ζc|+ e−α

et|ζe|+ |ζuu|))

, 0)

,

Puu+ (t)ζ =(

O(

e−αt)

ζuu, λO(

e−αt)

ζuu,O(

eαsst)

ζuu, ζuu)

.

For t ≤ 0 the projections P k−(t), k = c, e, ss, uu can be estimated by

P c−(t)ζ =(

ζc +O(

eαt)

ζss, 0, 0,O(

eαuut (|ζc|+ |ζss|)

)

)

,

P e−(t)ζ =(

0, ζe + λO(

eαt)

ζss, 0,O(

eαuut (|ζe|+ |λ| e−αtζss)

)

)

,

P ss− (t)ζ =(

O(

eαt)

ζss, λO(

e(αe+α)t

)

ζss, ζss,O(

eαuut)

ζss)

,

Puu− (t)ζ =(

0, 0, 0, ζuu +O(

eαuut (|ζc|+ |ζe|+ |ζss|)

)

)

.

Furthermore the projections are differentiable with respect to z = L,ϕ, ξe, µ, λ. The corresponding estimates possess

a similar structure as the estimate of the projections - merely the occurrences of ζk that are independent of z are

gone and the exponential rates αk are replaced by the (arbitrarily) slightly weaker rate αk, k = ss, e, uu. Furthermore

differentiating with respect to λ eliminates one order of λ in the estimates.

Before we give the proof of Lemma 7.5 we introduce the following definition:

We define for k = c, e, ss, uu and (ζc, ζe, ζss, ζuu) ∈ R4 the mappings

Pk : R4 → R, as Pk(ζc, ζe, ζss, ζuu) := ζk.

Proof of Lemma 7.5. The proof of Lemma 7.5 proceeds in the same way as the proof of [44, Lemma 2]. Since we

use some of the results later, we derive the components of the projections that differ from those in [44, Lemma 2]. A

straightforward calculation yields the idempotence and the commutativity of the projections. To derive the estimates

of the projections consider at first the transition matrices Φk±. Since Ass(vc, 0, 0, 0) = Auu(vc, 0, 0, 0) = 0, it follows

by Lemma 7.1 together with the estimates of Lemma 7.4 that

Φe+(t, τ)ζe = O(eα

e(t−τ))ζe, t ≥ τ ≥ 0, Φss− (t, τ)ζss = O(eαss(t−τ))ζss, 0≥ t ≥ τ,

Φss+ (t, τ)ζss = O(eαss(t−τ))ζss, t ≥ τ ≥ 0, Φuu− (t, τ)ζuu= O(eα

uu(t−τ))ζuu, 0≥ τ ≥ t. (7.32)

Φuu+ (t, τ)ζuu = O(eαuu(t−τ))ζuu, τ ≥ t ≥ 0,


We start with the calculation of the e-component of Puu+ (t), t ≥ 0. With Lemma 7.4 and (7.25), (7.27), (7.32) we get

Pe Puu+ (t)ζ = Xe+(t, 0, P

uu+ (0)Φ+(0, t)ζ)

= Xe+

(

t, 0, (0, 0, 0, Xuu(0, t, ζ)))

= λ

∫ t

0

Φe+(t, σ)[

DcAe(qc+(σ), q

e+(σ))(q

e+(σ))

2Xc+(σ, 0, (0, 0, 0, X

uu+ (0, t, ζ)))

+B(q+(σ))qss+ (σ)Xuu

+ (σ, 0, (0, 0, 0, Xuu+ (0, t, ζ)))

]

d σ

= λ

∫ t

0

Φe+(t, σ)[

DcAe(qc+(σ), q

e+(σ))(q

e+(σ))

2

∫ σ

0

Ac(q+(ω))qss+ (ω) Φuu(ω, t)ζuudω

+B(q+(σ))qss+ (σ) Φuu(σ, t)ζuu

]

d σ

= λ

∫ t

0

O(

eαe(t−σ)

)

[

λ2(

ξe∆eeαeσ +O

(

e2αeσ))2

∫ σ

0

(

∆sseαssω +O

(

e2αssω))

O(

eαuu(ω−t)

)

ζuudω

+(

∆sseαssσ +O

(

e2αssσ))

O(

eαuu(σ−t)

)

ζuu]

d σ (σ ≤ t)

= λO(e−αt)ζuu.

Similarly we derive the e-component of the P c+ projections:

Pe P c+(t)ζ = Xe+(t, 0, P

c+(0)Φ+(0, t)ζ)

= Xe+

(

t, 0, (Xc(0, t, ζ), 0, 0, 0))

= λ

∫ t

0

Φe+(t, σ)DcAe(qc+(σ), q

e+(σ))(q

e+(σ))

2Xc+(σ, 0, (X

c+(0, t, ζ), 0, 0, 0))d σ

= λ

∫ t

0


e+(σ))(q

e+(σ))

2[

ζc +

∫ 0

t

Ac(q+(ω))qss+ (ω) Φuu(ω, t)ζuudω

]

d σ (σ ≤ t)

= λ

∫ t

0

O(

eαe(t−σ)

)

λ2(

ξe∆eeαeσ +O

(

e2αeσ))2 [

ζc +

∫ 0

t

(

∆sseαssω +O

(

e2αssω))

O(

eαuu(ω−t)

)

ζuudω]

d σ

= λ2O(eαet)(

|ζc|+ e−αt|ζuu|)

and the e-component of the projections P ss+ :

Pe P ss+ (t)ζ = Xe+(t, 0, P

ss+ (0)Φ+(0, t)ζ) = Xe

+

(

t, 0, (0, 0, Xss(0, t, ζ), 0))

= 0.

Furthermore we obtain the c-component and the e-component of P e+:

Pc P e+(t)ζ = Xc+(t, 0, P

e+(0)Φ+(0, t)ζ) = Xc

+

(

t, 0, (0, Xe(0, t, ζ), 0, 0))

= 0,

Pe P e+(t)ζ = Xe+(t, 0, P

e+(0)Φ+(0, t)ζ)

= Xe+

(

t, 0, (0, Xe(0, t, ζ), 0, 0))

= Φe+(t, 0)[

Φe+(0, t)ζe + λ

∫ 0

t

Φe+(0, σ)[

DcAe(qc+(σ), q

e+(σ))(q

e+(σ))

2Xc+(σ, t, ζ)

+B(q+(σ)) qss+ (σ)Xuu

+ (σ, t, ζ)]

d σ]


= ζe + λ

∫ 0

t

Φe+(t, σ)[

DcAe(qc+(σ), q

e+(σ))(q

e+(σ))

2Xc+(σ, t, ζ) +B(q+(σ)) q

ss+ (σ)Xuu

+ (σ, t, ζ)]

d σ

= ζe + λ

∫ 0

t

O(

eαe(t−σ)

) [

λ2O(

e2αeσ)

Xc+(σ, t, ζ) +O

(

eαssσ)

Φuu(σ, t)ζuu]

d σ

= ζe + λ

∫ 0

t

O(

eαe(t−σ)

) [

λ2O(

e2αeσ)[

ζc +

∫ σ

t

Ac(q+(ω))qss+ (ω)Xuu

+ (ω, t, ζ)dω]

+O(

eαssσ)

O(

eαuu(ω−t)

)

ζuu]

d σ

= ζe + λ

∫ 0

t

O(

eαe(t−σ)

) [

λ2O(

e2αeσ)[

ζc +O(

eαssσ)

O(

eαuu(σ−t)

)

ζuu +O(

eαsst)

ζuu]

+O(

eαssσ)

O(

eαuu(σ−t)

)

ζuu]

d σ

= ζe + λ2O(

eαet)

ζc + λO(

e−αt)

ζuu.

Finally we derive the ss-component of the P e+ projections

Pss P e+(t)ζ = Xss+ (t, 0, P e+(0)Φ+(0, t)ζ)

= Xss+

(

t, 0, (0, Xe(0, t, ζ), 0, 0))

=

∫ t

0

Φss+ (t, σ)DeAss(q+(σ)) q

ss+ (σ)Xe

+(σ, 0, (0, Xe(0, t, ζ), 0, 0))dσ

=

∫ t

0

Φss+ (t, σ)DeAss(q+(σ)) q

ss+ (σ) Φe+(σ, 0)

[

Φe+(0, t)ζe

+ λ

∫ 0

t

Φe+(0, ω)[

DcAe(qc+(ω), q

e+(ω))(q

e+(ω))

2Xc+(ω, t, ζ) +B(q+(ω)) q

ss+ (ω)Xuu

+ (ω, t, ζ)]

dω]

dσ

=

∫ t

0

O(

eαss(t−σ)

)

O(

eαssσ) [

O(

eαe(σ−t)

)

ζe

+ λ

∫ 0

t

O(

eαe(σ−ω)

) [

λ2O(

e2αeω)

[ζc +

∫ ω

t

Ac(q+())qss+ ()Xuu+ (, t, ζ)d]

+O(

eαssω)

O(

eαuu(ω−t)

)

ζuu]

dω]

dσ

= O(

e(αss−αe)t

)

ζe + λO(

eαsst)

∫ t

0

∫ 0

t

λ2O(

eαe(σ+ω)

)

[ζc +

∫ ω

t

O(

eαss)

O(

eαuu(−t)

)

ζuud]

+O(

eαe(σ−ω)

)

O(

eαssω)

O(

eαuu(ω−t)

)

ζuudωdσ

= O(

e(αss−αe)t

)

ζe + λ2O(

eαsst)

ζc + λO(

e−αet)

O(

e(αss−α)t

)

ζuu,

as well as, the ss-component of the P ss+ projections

Pss P ss+ (t)ζ = Xss+ (t, 0, P ss+ (0)Φ+(0, t)ζ)

= Xss+

(

t, 0, (0, 0, Xss(0, t, ζ), 0))

= Φss+ (t, 0)[

Φss+ (0, t)ζss +

∫ 0

t

Φss+ (0, σ)(

DuuAss(q+(σ)) q

ss+ (σ)Xuu

+ (σ, t, ζ)

+DeAss(q+(σ)) q

ss+ (σ)Xe

+(σ, t, ζ) +DcAss(q+(σ)) q

ss+ (σ)Xc

+(σ, t, ζ))

dσ]

= ζss +O(

eαsst)

∫ 0

t

Φuu+ (σ, t)ζuu + Φe+(σ, t)ζe + λ

∫ σ

t

Φe+(σ, ω)[

DcAe(qc+(ω), q

e+(ω))(q

e+(ω))

2Xc+(ω, t, ζ)


+B(q+(ω)) qss+ (ω)Xuu

+ (ω, t, ζ)]

dω + ζc +

∫ σ

t

Ac(q+(ω))qss+ (ω)Xuu

+ (ω, t, ζ)dωdσ

= ζss +O(

eαsst)

ζuu +O(

eαsst)

∫ 0

t

O(

eαe(σ−t)

)

ζe

+ λ

∫ σ

t

O(

eαe(σ+ω)

) [

[ζc +

∫ ω

t

O(

eαss)

O(

eαuu(−t)

)

ζuud]

+O(

eαssω)

O(

eαuu(ω−t)

)

ζuu]

dω + ζc +

∫ σ

t

O(

eαssω)

O(

eαuu(ω−t)

)

ζuudωdσ

= ζss +O(

eαsst)

ζuu +O(

e(αss−αe)t

)

ζe +O(

eαsst)

ζc.

The remaining projections are estimated in the same way. For the calculation of the Projections P k−, k = c, e, ss, uu

we recall that qe− ≡ 0.

The differentiability of the projections with respect to the variables z = L,ϕ, ξe, µ, λ follows from the fact that the

right hand sides of the variational equations (7.23), (7.29) are smooth in these variables. By definition

Pjd

dz

[

Φ+(t, τ)Pk+(τ)

]

ζ =d

dzXj

+(t, τ,Φ+(τ, 0)Pk+(0)Φ+(0, τ)ζ), j, k = c, e, ss, uu.

Plugging in the solutions X+, see (7.26) - (7.25), we obtain forall Pe ddz[

Φ+(t, τ)Pc+(τ)

]

ζ the following expression

d

dz

(

λ

∫ t

0


e+(σ))(q

e+(σ))

2[

ζc +

∫ 0

τ

Ac(q+(ω))qss+ (ω) Φuu(ω, τ)ζuudω

]

d σ)

,

where we recall that q+ depends on ϕ, ξe. To estimates this expression we need to differentiate the transition matrices

Φuu+ , and Φe+ of (7.24u) and (7.24e). As in the proof of Lemma 7.4 we differentiate (7.24u) and (7.24e). According to

the estimates of q+ in Lemma 7.4 and (7.32) we may apply Corollary 7.3 to derive:

d

dzΦuu+ (t, τ)ζuu = O(eα

uu(t−τ))ζuud

dzΦe+(t, τ)ζ

e = O(eαe(t−τ))ζe, t ≤ τ, (7.33)

where we have used slightly decreased rates αuu and αe. With (7.33) we estimate ddz Φ

uu+ (t, τ)ζuu as the corresponding

projections before. The other differentiated projections can be estimated similarly.

Corollary 7.6. If τ ≥ t ≥ 0 and z = L,ϕ, ξe, µ, then

d

dz[Φ+(t, τ)P

c+(τ)]ζ =

(

O(

e−αt)

ζuu, λ2O(

eαe

(|ζc|+ e−αt|ζuu|))

,O(

eαssτ (|ζc|+ e−αt|ζuu|)

)

, 0

)

,

d

dz[Φ+(t, τ)P

e+(τ)]ζ =

(

0,O(

eαeτ (λ2|ζc|+ e−α

et|ζe|+ λe−(α+αe)tζuu))

,

O(

eαssτ (λ2|ζc|+ e−α

et|ζe|+ λ eαe(τ−t)e−αt|ζuu|)

)

, 0

)

,

d

dz[Φ+(t, τ)P

ss+ (τ)]ζ =

(

0, 0,O(

eαssτ (|ζc|+ e−α

et|ζe|+ e−αsst|ζss|+ |ζuu|)

)

, 0

)

,

d

dz[Φ+(t, τ)P

uu+ (τ)]ζ=

(

O(

e−αuut[eβ

suτ + 1])

ζuu, λO(

e−αuut[eβ

suτ + 1])

ζuu,

O(

e−αuut[eβ

suτ + eαssτ ])

ζuu,O(

eαuu(τ−t)

)

ζuu)

,


where βsu := αuu + αss. If τ ≤ t ≤ 0, then the derivative ddz [Φ−(t, τ)P

k−(τ)]ζ =: Pk−(t, τ)ζ, k = c, e, ss, uu can be

estimated by:

d

dz[Φ−(t, τ)P

c−(τ)]ζ =

(

O(

eαt)

ζss, 0, 0,O(

eαuuτ (|ζc|+ eαt|ζss|)

)

,

)

,

d

dz[Φ−(t, τ)P

e−(τ)]ζ =

(

0,O(

eαeτ (e−α

et|ζe|+ λe(α−αe)tζss)

)

, 0,O(

eαuuτ (eα

e(τ−t)|ζe|+ λ eαe(τ−t)eαt|ζss|)

)

,

)

,

d

dz[Φ−(t, τ)P

ss− (τ)]ζ =

(

O(

e−αsst[eβ

suτ + 1])

ζss, λO(

e−αsst[eβ

suτ + 1])

ζss,

O(

e−αss(τ−t)

)

ζuu,O(

e−αsst[eβ

suτ + eαuuτ ]

)

ζss)

,

d

dz[Φ−(t, τ)P

uu− (τ)]ζ=

(

0, 0,O(

eαuuτ (|ζc|+ |ζe|+ |ζss|+ e−α

uut|ζuu|))

, 0

)

.

If z = λ, then the same estimates apply, with the exception that whenever λ occurs in the estimate, then the order of

that term is reduced by one.

Proof of Corollary 7.6. To prove Corollary 7.6 we differentiate Φ±(t, τ)Pk±(τ)ζ by exploiting the definition of Φ±

and P k±. The resulting terms are estimated as in Lemma 7.5, where we use in particular (7.33).

7.4 The solutions of the inhomogeneous equations

In this section we show that equation (7.7),

Y±(t) = Df(

q±(t))

· Y±(t) + g±(t),

with the initial/boundary conditions (7.8) and the coupling equation (7.9) possesses a unique solution Y±. The

functions g± := (gc±, λ ge±, g

ss± , g

uu± ), where (gc±, g

e±, g

ss± , g

uu± ) belongs to the normed function spaces

(

V±, ‖ · ‖V±

)

,

defined below. The factor λ in the definition of g± is due to the fact that the e-component of the inhomogeneity h

in (7.6) vanishes, if λ = 0. For the sake of brevity we write g± ∈ V± instead of (gc±, ge±, g

ss± , g

uu± ) ∈ V±. Denote by

C0(I±,R) the space of continuous functions mapping the interval I± into R and recall that ν is some arbitrarily small,

but positive constant. We define

V c+ :=(

C0(I+,R), ‖ · ‖c+

)

, with ‖h‖c+ := supt∈I+ eνt |h(t)| ,

V e+ :=(

C0(I+,R), ‖ · ‖e+

)

, with ‖h‖e+ := supt∈I+

e−αet |h(t)|

,

V ss+ :=(

C0(I+,R), ‖ · ‖ss+

)

, with ‖h‖ss+ := supt∈I+

e−αsst |h(t)|

,

V uu+ :=(

C0(I+,R), ‖ · ‖uu+

)

, with ‖h‖uu+ := supt∈I+

e−αuu(t−L) |h(t)|

,

V c− :=(

C0(I−,R), ‖ · ‖c−

)

, with ‖h‖c− := supt∈I− e−νt |h(t)| ,

V e− :=(

C0(I−,R), ‖ · ‖e−

)

, with ‖h‖e− := supt∈I−

e−αe(t+L) |h(t)|

,

V ss− :=(

C0(I−,R), ‖ · ‖ss−

)

, with ‖h‖ss− := supt∈I−

e−αss(t+L) |h(t)|

,

V uu− :=(

C0(I−,R), ‖ · ‖uu−

)

, with ‖h‖uu− := supt∈I−

e−αuut |h(t)|

,

7.4 The solutions of the inhomogeneous equations 115

where we recall the definition of I+ := [0, L] and I− := [−L, 0]. Further we recall that αss, αe, αuu denote slightly

weaker (exponential) rates than αss, αe, αuu. Next we introduce the function spaces V±:

V+ :=

h = (hc, he, hss, huu) ∈ C0(I+,R4) | hk ∈ V k+ , k = c, e, ss, uu

and ‖h‖+ := ‖hc‖c+ + ‖h‖e+ + ‖h‖ss+ + ‖h‖uu+ ,

V− :=

h = (hc, he, hss, huu) ∈ C0(I−,R4) | hk ∈ V k+ , k = c, e, ss, uu

and ‖h‖− := ‖hc‖c− + ‖h‖e− + ‖h‖ss− + ‖h‖uu− .

Finally we define the space V with the norm ‖ · ‖ by

V := V+ × V− and ‖(h+, h−)‖V := ‖h+‖+ + ‖h−‖−.

Furthermore, for h± = (hc, he, hss, huu) ∈ V± and k = c, e, ss, uu we define the norms

‖hk±‖∞ := supt∈I±

|hk±(t)|

and ‖h±‖∞ := supt∈I±

|hc±(t)|+ |he±(t)|+ |h

ss± (t)|+ |huu± (t)|

.

Next we define functions Y± that solve Equation (7.7). Let a := (ac, ae, ass, auu) ∈ R4 be some vector. We introduce

ac(auu) := −PcPuu+ (L)(0, 0, 0, auu) =

∫ 0

L

Ac(q+(σ))qss+ (σ)Φuu+ (σ, 0)auu dσ. (7.34)

We define

Y+(t, a) := Φ+(t, L)Pc+(L)(a

c(auu), 0, 0, 0) + Φ+(t, L)Puu+ (L)(0, 0, 0, auu)

+

∫ t

L

Φ+(t, σ)[

P c+(σ) + Puu+ (σ)]

g+(σ) dσ +

∫ t

0

Φ+(t, σ)[

P e+(σ) + P ss+ (σ)]

g+(σ) dσ, t ∈ I+,

Y−(t, a) := Φ−(t,−L)Pc−(−L)(a

c, 0, 0, 0) + Φ−(t,−L)Pe−(−L)(0, a

e, 0, 0) + Φ−(t,−L)Pss− (−L)(0, 0, ass, 0)

+

∫ t

−L

Φ−(t, σ)[

P c−(σ) + P e−(σ) + P ss− (σ)]

g−(σ) dσ +

∫ t

0

Φ−(t, σ)Puu− (σ)g−(σ) dσ, t ∈ I−. (7.35)

The definition of ac(auu) is due to the initial/boundary conditions (7.8) (i). In fact a

c(auu) is chosen such that

PcY+(0, a) = 0. In contrast to [44] we stipulate the c-component of the solution of Theorem 2.4 not at the boundary

of the interval [−L,L], but in the middle, vc(0) = ϕ. Therefore we need the correction term ac(auu) here. Further we

remark that the term Y+(·, a) depends only on auu and similarly Y−(·, a) depends only on ac, ae, ass.

Lemma 7.7. For all a ∈ R4 and g± ∈ V± the functions Y+(·, a) and Y−(·, a) are solutions of the inhomogeneous

differential equations

Y±(t, a) = Df(

q±(t))

· Y±(t, a) + g±(t).

Moreover Y+(·, a) and Y−(·, a) depend linearly on (a, g±) and satisfy the boundary conditions:

Y c+(L, a) := Pc Y+(L, a) = 0, Y ss+ (0, a) := Pss Y+(0, a) = 0,

Y e+(0, a) := Pe Y+(0, a) = 0, Y uu− (0, a) := Puu Y−(0, a) = 0.


Proof of Lemma 7.7. Recall that Φ+(·, ·) is the transition matrix of the variational equation (7.23). Hence

Φ+(t, L)Pc+(L)(a

c, 0, 0, 0) + Φ+(t, L)Puu+ (L)(0, 0, 0, auu)

is a solution of the linear equation X+(t) = Df(

q+(t))

·X+(t). Further a calculation shows that

d

dt

(

∫ t

L

Φ+(t, σ)[

P c+(σ) + Puu+ (σ)]

g+(σ) dσ +

∫ t

0

Φ+(t, σ)[

P e+(σ) + P ss+ (σ)]

g+(σ) dσ)

= Df(q+(t))(

∫ t

L

Φ+(t, σ)[

P c+(σ) + Puu+ (σ)]

g+(σ) dσ +

∫ t

0

Φ+(t, σ)[

P e+(σ) + P ss+ (σ)]

g+(σ) dσ)

+ g+(t).

Therefore Y+(·, a) is a solution of Y+(t, a) = Df(

q+(t))

· Y+(t, a) + g+(t). The linearity in (a, g+) is obvious, since

ac(auu) is linear in auu. The same applies to Y−(·, a). Regrading the boundary values of Y±(·, a) we find

Y+(0, a) = P c+(0)Φ+(0, L)(ac(auu), 0, 0, 0) + Puu+ (0)Φ+(0, L)(0, 0, 0, a

uu) + (0, ξe, 0, 0)

+[

P c+(−L) + Puu+ (−L)]

∫ 0

L

Φ+(0, σ)g+(σ) dσ,

Y−(0, a) = P ss− (0)Φ−(0,−L)(0, 0, ass, 0) + P e−(0)Φ−(0,−L)(0, a

e, 0, 0) + P c−(0)Φ−(0,−L)(ac, 0, 0, 0)

+[

P c−(0) + P e−(0) + P ss− (0)]

∫ 0

−L

Φ−(0, σ)g−(σ) dσ,

where we exploited the commutativity of the projections with the transition matrices Φ±. Inspecting the definition

of the projections, (7.31), it is immediate that Y±(0, a) are zero in the claimed components. Regarding Y c+(L, a), we

recall that according to the definition of ac, the c-components of the terms Puu+ (L)(0, 0, 0, auu) and P c+(L)(ac, 0, 0, 0)

cancel each other. Hence Y c+(L, a) = 0.

So far we have shown that Y± do satisfy the initial/boundary conditions (7.8). In what follows we show that by

choosing the vector (ac, ae, ass, auu) appropriately the solutions Y± satisfy the coupling condition (7.9).

Lemma 7.8. For any d = (0, de, dss, duu) ∈ R4 and all (g−, g+) ∈ V , there exists an L0 such that for all L > L0

there is a unique vector a = (ac, ae, ass, auu) such that the coupling condition (7.9),

Y+(L, a)− Y−(−L, a) = d,

is satisfied.

Proof of Lemma 7.8. The proof is based on the ideas of [44, Lemma 3]. At first we exploit (7.32) and Lemma 7.4

to estimate

ac(auu) =

∫ 0

L

Ac(q+(σ))qss+ (σ)Φuu+ (σ, 0)auu dσ = O

(

e−αLauu)

. (7.36)

Once more we recall that we have chosen ac so that c-components of the terms Puu+ (L)(0, 0, 0, auu) and P c+(L)(a

c, 0, 0, 0)

cancel each other. Next we exploit (7.36), the form of the projections given in Lemma 7.5 and the representations of

the solutions of the variational equations, (7.25) - (7.28) and (7.30), to estimate


Y+(L, a) =(

0, λO(

e−αL)

auu,O(

eαssL)

auu, auu)

+[

P e+(L) + P ss+ (L)]

∫ L

0

Φ+(L, σ)g+(σ) dσ,

Y−(−L, a) =(

ac +O(

e−αL)

ass, ae + λO(

e−(αe+α)L)

ass, ass,O(

e−αuuL)

[|ac|+ |ae|+ |ass|])

+ Puu− (−L)

∫ −L

0

Φ−(−L, σ)g−(σ) dσ.

With these estimates Equation (7.9) can be written as

N · a+ y = d,

where N is given by

N =

−1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 1

+

0 0 O(

e−αL)

0

0 0 λO(

e−L)

λO(

e−αL)

0 0 0 O(

eαssL)

O(

e−αuuL)

O(

e−αuuL)

O(

e−αuuL)

0

and

y =[

P e+(L) + P ss+ (L)]

∫ L

0

Φ+(0, σ)g+(σ) dσ − Puu− (−L)

∫ −L

0

Φ−(0, σ)g−(σ) dσ. (7.37)

Clearly the matrix N is invertible, since it is a small perturbation of identity matrix. Hence a := N−1(d − y) is well

defined and solves (7.9).

In accordance with the above considerations we find that a = a(d, ϕ, ξe, δ, µ, λ, g), where g := (g+, g−). For the sake

of brevity we suppress the dependency on ϕ, ξe, δ, µ, λ, in our notation and write a = a(d, g). Further note that (7.35)

yields a solution of (7.7) that satisfies the initial/boundary condition and the coupling condition (7.9), for fixed d ∈ R4.

By means of Y+(·, a(d, g)), Y−(·, a(d, g)) we reformulate the nonlinear equation (7.6) into a fixed point problem. To

this end we define (for fixed L > L0 and any ϕ, ξe, δ, µ, λ) the operator NL : R4 × V → V that assigns (d, g) the

solutions (Y+(·, a), Y−(·, a)):

NL : (d, g) 7→ (Y+(·, a), Y−(·, a)) and a = a(d, g),

where we recall that a depends also on ϕ, ξe, δ, µ, λ. Further observe that operator NL is linear in (d, g). Since Y±(·, a)

depend smoothly on d, ϕ, ξe, δ, µ, λ, the operator NL depends smoothly on these quantities. Analogously to [44, Lemma

4] we state the following lemma:

Lemma 7.9. If L is sufficiently large and δ is sufficiently small, then

(i) NL maps into V and ‖NL‖V is uniformly bounded in L.

(ii) NL is smooth in ϕ, ξe, µ, λ, d and ‖DzNL‖V , z = ϕ, ξe, µ, λ, d is uniformly bounded in L.

(iii) NL is (Frechet-) differentiable with respect to g ∈ V and ‖DgNL‖L(V ) is uniformly bounded in L.

Also higher derivatives of NL are uniformly bounded in L.


Proof of Lemma 7.9. The proof of Lemma 7.9 proceeds as in [44]. Since it contains crucial estimates for the further

considerations we present it here in detail. We start with showing (i). At first we note that Y±(·, a) are continuous

functions, which is immediate by inspecting (7.35) and the definition of ac. To show that NL maps into V , it is enough

to show the uniform boundedness of ‖NL‖V . That means we have to find a constant CN , that is independent of L,

such that

‖NL‖V = ‖Y+(·, a), Y−(·, a)‖V ≤ CN ,

where a is given by a = N−1(d− y). Thus we need to estimate the solutions Y±(·, a). We start with the integral terms

of Y+(·, a):

∫ t

L

Φ+(t, σ)Pc+(σ)g+(σ) dσ

=

∫ t

L

Φ+(t, σ)(

gc+(σ) +O(

e−ασ)

guu+ (σ), λ2O(

eαeσ) (

|gc+(σ)|+ e−ασ|guu+ (σ)|)

,O(

eαssσ) (

|gc+(σ)|+ |guu+ (σ)|

)

, 0)

dσ,

where we exploited the estimates of Lemma 7.5. The above line can be considered as the integration of a solution of the

variational equation (7.23) with the expression in the brackets as initial value. According to (7.25), the uu-component

of the above integral is zero. Exploiting the representations (7.26), (7.27) and (7.28) we find that for t ∈ I+ holds

Pc∫ t

L

Φ+(t, σ)Pc+(σ)g+(σ) dσ =

∫ t

L

gc+(σ) +O(

e−ασ)

guu+ (σ) dσ

=

∫ t

L

e−νσ‖gc+‖c+ +O

(

e−ασ)

eαuu(σ−L)‖guu+ ‖

uu+ dσ = O

(

e−νt)

‖gc+‖c+ +O

(

e−αt)

‖guu+ ‖uu+ ,

Pe∫ t

L


=

∫ t

L

Φe+(t, σ)[

λ2O(

eαeσ) (

|gc+(σ)|+ e−ασ|guu+ (σ)|)]

+ λ

∫ t

σ

Φe+(t, ω)DcAe(qc+(ω), q

e+(ω)) (q

e+(ω))

2[

gc+(σ) +O(

e−ασ)

guu+ (σ)]

dω d σ

=

∫ t

L

O(

eαe(t−σ)

)[

λ2O(

eαeσ) (


(

e−ασ)


uu+

)]

+ λ2∫ t

σ

O(

eαe(t+ω)

) [


(

e−ασ)


uu+

]

dω dσ

= λO(

eαet) (

‖gc+‖c+ +O

(

e−αt)

‖guu+ ‖uu+

)

,

Pss∫ t

L


=

∫ t

L

Φss+ (t, σ)O(

eαssσ) (

|gc+(σ)|+ |guu+ (σ)|

)

+

∫ t

σ

Φss+ (t, ω)(

DcAss(q+(ω)) q

ss+ (ω)

[

gc+(σ) +O(

e−ασ)

guu+ (σ)]

+DeAss(q+(ω)) q

ss+ (ω)

(

Φe+(ω, σ)[

λ2O(

eαeσ) (

|gc+(σ)|+ e−ασ|guu+ (σ)|)]

+ λ

∫ ω

σ

Φe+(t, τ)DcAe(qc+(ω), q

e+(ω)) (q

e+(ω))

2[

gc+(σ) +O(

e−ασ)

guu+ (σ)]

d τ)

)

dτ dω dσ


= O(

eαsst)

∫ t

L

e−νσ‖gc+‖c+ + eα

uu(σ−L)‖guu+ ‖uu+ dσ

+O(

eαsst)

∫ t

L

∫ t

σ


(

e−ασ)


uu+ dω dσ

+ λ2O(

eαsst)

∫ t

L

∫ t

σ

O(

eαeω) (


(

e−ασ)


uu+

)

dω dσ

+ λO(

eαsst)

∫ t

L

∫ t

σ

∫ ω

σ

O(

eαe(t+τ)

) [


(

e−ασ)


uu+

]

d τ dω dσ

= O(

eαsst) (

‖gc+‖c+ + ‖guu+ ‖

uu+

)

,

Puu∫ t

L

Φ+(t, σ)Pc+(σ)g+(σ) dσ = 0.

Recall that g± = (gc±, λ ge±, g

ss± , g

uu± ). Exploiting further that αss+ αe−αe ≤ αss we estimate analogously (for t ∈ I+):

∫ t

0

Φ+(t, σ)Pe+(σ)g+(σ) dσ

=

(

0, λO(

eαet)

‖gc+‖c+ +O

(

eαet)

‖ge+‖e+ + λO

(

e−αL)

‖guu+ ‖uu+ , λO

(

eαsst)[

λ ‖gc+‖c+ + ‖ge+‖

e+ + ‖guu+ ‖

uu+

]

, 0

)

, (7.38e)

∫ t

0

Φ+(t, σ)Pss+ (σ)g+(σ) dσ =

(

0, 0,O(

eαsst)[

‖gc+‖c+ + λ ‖ge+‖

e+ + ‖gss+ ‖

ss+ + ‖guu+ ‖

uu+

]

, 0

)

, (7.38s)

∫ t

L

Φ+(t, σ)Puu+ (σ)g+(σ) dσ

=

(

O(

e−αt)

‖guu+ ‖uu+ , λO

(

e−αt)

‖guu+ ‖uu+ ,O

(

eαsst)

‖guu+ ‖uu+ ,O

(

eαuu(t−L)

)

‖guu+ ‖uu+

)

. (7.38u)

And for t ∈ I− we find

∫ t

−L

Φ−(t, σ)Pc−(σ)g−(σ) dσ

=

(

O(

eνt)

‖gc−‖c− +O

(

e−αL)

‖gss− ‖ss− , 0, 0,O

(

eαuut)[

‖gc−‖c− + ‖gss− ‖

ss−

]

)

, (7.39c)

∫ t

−L

Φ−(t, σ)Pe−(σ)g−(σ) dσ

=

(

0,O(

eαe(t+L)

)

‖ge−‖e− + λO

(

e−αL)

‖gss− ‖ss− , 0, λO

(

eαuut)[

‖ge−‖e− + ‖gss− ‖

ss−

]

)

, (7.39e)

∫ t

−L

Φ−(t, σ)Pss− (σ)g−(σ) dσ

=

(

O(

e−αL)

‖gss− ‖ss− , λO

(

eαet)

O(

e−αL)

‖gss− ‖ss− ,O

(

eαss(t+L)

)


(

eαuut)

‖gss− ‖ss−

)

, (7.39s)

∫ t

0

Φ−(t, σ)Puu− (σ)g−(σ) dσ =

(

0, 0, 0,O(

eαuut)[

‖gc−‖c− + λ ‖ge−‖

e− + ‖gss− ‖

ss− + ‖guu− ‖

uu−

]

)

. (7.39u)


Finally we calculate more thoroughly

Φ+(t, L)Puu+ (L)(0, 0, 0, ζuu)

= Φ+(t, L)Φ+(L, 0)Puu+ (0)Φ+(0, L)(0, 0, 0, ζ

uu)

= Φ+(t, 0)(0, 0, 0, Φuu+ (0, L)ζuu)

=(

O(

e−αteαuu(t−L)

)

ζuu, λO(

e−αteαuu(t−L)

)

ζuu,O(

eαssteα

uu(t−L))

ζuu,O(

eαuu(t−L)

)

ζuu)

,

where we exploited the definition of Puu+ , (7.25)–(7.28) and (7.32). With these estimates and Lemma 7.5 we find that

Y+(t, a) =

(

O(

e−αteαuu(t−L)

)

auu, λO(

e−αteαuu(t−L)

)

auu,O(

eαssteα

uu(t−L))

auu,O(

eαuu(t−L)

)

auu)

+

(

ac, 0,O

(

eαsst

ac)

0

)

+

(

O(

e−νt)

‖gc+‖c+ +O

(

e−αt)

‖guu+ ‖uu+ ,

O(

eαet)(

λ‖gc+‖c+ + ‖ge+‖

e+ + λ e−αL‖guu+ ‖

uu+

)

,O(

eαsst)

‖g+‖+,O(

eαuu(t−L)

)

‖guu+ ‖uu+

)

.

Recall that in (7.36) we have seen that ac(auu) = O(

e−αLauu)

. This finally yields

Y+(t, a) =

(

O(

e−αL)

auu, λO(

e−αL)

auu,O(

eαsst)

auu,O(

eαuu(t−L)

)

auu)

+

(

O(

e−νt)

‖gc+‖c+ +O

(

e−αt)

‖guu+ ‖uu+ ,

O(

eαet)(

λ‖gc+‖c+ + ‖ge+‖

e+ + λ e−αL‖guu+ ‖

uu+

)

,O(

eαsst)

‖g+‖+,O(

eαuu(t−L)

)

‖guu+ ‖uu+

)

.

For Y−(t, a) we estimate similarly

Y−(t, a) =

(

0, eαe(t+L)ae, 0,O

(

eαuut)

ae)

+

(

ac, 0, 0,O(

eαuut)

ac)

+

(

O(

e−αL)

ass, λO(

e−αL)

ass,O(

eαss(t+L)

)

ass,O(

eαuut)

ass)

+

(

O(

eνt)

‖gc−‖c− +O

(

e−αL)


(

eαe(t+L)

)

‖ge−‖e− + λO

(

eαet)

O(

e−αL)

‖gss− ‖ss− ,

O(

eαss(t+L)

)


(

eαuut)

‖g−‖−

)

.

Hence ‖Y±(·, a)‖± are bounded in L, if ass, auu are bounded in L and ac = O (eνt) , t ∈ I−. To obtain these estimates

we recall that a = N−1(d − y). Since N is for sufficient large L close to the identity matrix, we see that N−1 exists.

A calculation shows that N−1 is given by:

N−1 =

−1 +O(

e(αss−αuu−α)L

)

O(

e(αss−αuu−α)L

)

O(

e−αL)

O(

e(αss−α)L

)

λO(

e(αuu−α)L

)

−1 + λO(

e(αuu−α)L

)

λO(

e−L)

λO(

e−αL)

O(

e(αss−αuu)L

)

O(

e(αss−αuu)L

)

−1 +O(

e(αss−αuu)L

)

O(

eαssL)

O(

e−αuuL)

O(

e−αuuL)

O(

e−αuuL)

1 +O(

e−(αuu+α)L)

.


Moreover (7.37) and the estimates (7.38) and (7.39) show that

y =(

0, λO(

eαeL)

‖ge+‖e+ + λO

(

eαeL)

‖gc+‖c+ + λ O

(

e−αL)

‖guu+ ‖uu+ , O

(

eαssL)

‖g+‖+, O(

e−αuuL)

‖g−‖−)

.

Thus

a =

O(

e−αL)

dss +O(

e(αss−α)L

)

‖d‖∞

−de + λO(

e−L)

‖d‖∞

−dss +O(

eαssL)

‖d‖∞

duu +O(

e−αuuL)

‖d‖∞

+

O(

e(αss−α)L

)

‖g‖V

λ[

O(

eαeL)

‖ge+‖e+ +O

(

eαeL)

‖gc+‖c+ +O

(

e−αL)

‖g‖V]

O(

eαssL)

‖g‖V

O(

e−αuuL)

‖g‖V

(7.40)

where we recall that d = (0, de, dss, duu). The first line of Equation (7.40) shows that ac = O(

e−αL)

, if ‖d‖∞, ‖g+, g−‖V

are bounded. Consequently ‖NL‖V is uniformly bounded in L, for L > L0.

Next we consider (ii). Let (d, ϕ, ξe, δ, µ, λ, g) be fixed and denote by L(R, V ) the space of the linear (bounded) mappings

from R into V . Then DzNL(d, g)(·) ∈ L(R, V ), for z = ϕ, ξe, µ, λ, dk, k = e, ss, uu. To show that DzNL exists, we

differentiate NL(d, g)(·) = (Y+(·, a), Y−(·, a)) and show that the resulting map ( ddzY+(·, a),ddzY−(·, a)) is indeed the

derivative of NL. Regarding the uniformly boundedness, we consider the map

(d, ϕ, ξe, δ, µ, λ, g) 7→ DzNL(d, g)(·) = (d

dzY+(·, a),

d

dzY−(·, a)), z = ϕ, ξe, µ, λ, dk, k = e, ss, uu.

Hence we have to show that there is a constant C ′N > 0, independent of L, such that for all L > L0:

∥

∥DzNL(d, g)(·)∥

∥

V=∥

∥(d

dzY+(·, a),

d

dzY−(·, a))

∥

∥

V≤ C ′

N .

Since ddzY±(·, a) = DzY±(·, a) + DaY±(·, a) · Dza, we need estimates of Dza = (Dza

c, Dzae, Dza

ss, Dzauu). These

estimates can be obtained by differentiating (7.9) with respect to z and exploiting Corollary (7.6). The procedure is

rather technical, but straightforward. Therefore we omit the details here.

Finally we show (iii). Note at first that DgNL(d, g)(·) is an element of L(V ) for any fixed (d, ϕ, ξe, δ, λ, g). To show

that the derivative exists we observe that NL is affine in g and that the linear part of NL is given by the operator

DgNL(d, g) : V → V

(h+, h−) 7→(

∫ (·)

0

Φ+(·, σ)[Pc+(σ) + P e+(σ) + Puu+ (σ)]h+(σ) dσ +

∫ (·)

−L

Φ+(·, σ)Pss+ (σ)h+(σ) dσ,

∫ (·)

0

Φ−(·, σ)[Pc−(σ) + P e−(σ) + P ss− (σ)]h−(σ) dσ +

∫ (·)

L

Φ−(·, σ)Puu− (σ)h−(σ) dσ

)

It is immediate that DgNL is indeed a linear operator in (h+, h−) ∈ V . Furthermore the estimates of the integral

terms above show that DgNL maps into V and that DgNL is a bounded operator. Consequently DgNL is indeed the

derivative of NL. It remains to show the uniformly boundedness of DgNL, i.e. to show that

∥

∥DgNL(d, g)(·)∥

∥

V≤ CN ,

for some CN > 0 and all (d, ϕ, ξe, δ, µ, λ, g). But this follows by the estimates of the integral terms of NL above.

Showing the uniformly boundedness of higher derivatives of NL proceeds similarly.


7.5 The fixed point equation

In this section we rewrite the boundary value problem (7.6), (7.4) together with the coupling condition (7.5) into a

fixed point equation and solve this equation. To this end we recall the definition of the space V on page 114 and define

the Nemitskii operators

G± : V± × R6 × Jµ × Jλ → g | g : I± → R4,

(X±, ϕ, ξe, δ, µ, λ) 7→ f

(

q± +X±

)

− f(

q±)

−Df(

q±)

·X±.

We observe that G± depend also (indirectly) on L, since the spaces V depend on L. To simplify our notation we write

G±(X±) = G±(X±, ϕ, ξe, δ, µ, λ). The next Lemma shows that G± maps V± × R6 × Jµ × Jλ into V±. Hence we may

define the operator

G±(d, ϕ, ξe, δ, µ, λ,X+, X−) := (X+, X−)−NL(d,G+(X+), G−(X−)). (7.41)

Supposed it exists, NL(d,G+(·), G−(·)) maps onto a solution X± of (7.6):

X±(t) = Df(q±(t))X±(t) + h(t,X±),

which satisfies the boundary conditions (7.4). Moreover, for this solution holds true X+(0)−X−(0) = d. Thus (7.6)

with (7.4) and (7.5) is equivalent to

G±(d, ϕ, ξe, δ, µ, λ,X+, X−) = 0. (7.42)

We solve this equation by means of the Implicit Function Theorem for operators [12, Theorem 2.3]. But at first we

prove the following result that corresponds to [44, Lemma 5].

Lemma 7.10. There exist neighbourhoods of zero UV±⊂ V± such that the operators G± map UV±

× R6 × Jµ × Jλ

into V±. The operators G± are smooth and the following estimates hold true uniformly in L:

Gc+(X+) = O(

|Xuu+ | |X

ss+ + qss+ |

)

, Gc−(X−) = O(

|Xss− | |X

uu− + quu− |

)

,

Ge+(X+) = λO(

|Xe+ + qe+| |X

e+|+ |X

uu+ | |X

ss+ + qss+ |

)

, Ge−(X−) = λO(

|Xe−|

2 + |Xss− | |X

uu− + quu− |

)

,

Gss+ (X+) = O(

|qss+ | |X+|2 + |X+| |X

ss+ |)

, Gss− (X−) = O(

|X−| |Xss− |)

,

Guu+ (X+) = O(

|X+| |Xuu+ |)

, Guu− (X−) = O(

|quu− | |X−|2 + |X−| |X

uu− |)

.

Furthermore, similar estimates hold true for DzGk±(X±), k = c, e, ss, uu, z = ϕ, ξe, µ, λ, where any occurrence of

qss/e/uu± in the above estimates is replaced by the sum q

ss/e/uu± +Dzq

ss/e/uu± . Moreover

DXcGc+(X+) = O(

|Xuu+ | |X

ss+ + qss+ |

)

, DXcGe+(X+) = λO(

|Xe+| |X

c+|+ |X

uu+ | |X

ss+ + qss+ |

)

,

DXeGc+(X+) = O(

|Xuu+ | |X

ss+ + qss+ |

)

, DXeGe+(X+) = λO(

|Xe+|+ |X

uu+ | |X

ss+ + qss+ |

)

,

DXssGc+(X+) = O(

|Xuu+ |)

, DXssGe+(X+) = λO(

|Xuu+ |)

,

DXuuGc+(X+) = O(

|Xuu+ | |X

ss+ + qss+ |+ |X

ss+ |)

, DXuuGe+(X+) = λO(

|X+| |Xss+ + qss+ |

)

,

7.5 The fixed point equation 123

DXcGss+ (X+) = O(

|Xss+ |)

, DXcGuu+ (X+) = O(

|Xuu+ |)

,

DXeGss+ (X+) = O(

|Xss+ |)

, DXeGuu+ (X+) = O(

|Xuu+ |)

,

DXssGss+ (X+) = O(

|X+| |Xss+ + qss+ |

)

, DXssGuu+ (X+) = O(

|Xuu+ |)

,

DXuuGss+ (X+) = O(

|Xss+ |)

, DXuuGuu+ (X+) = O (|X+|) .

Analogous estimates hold true for DXG−.

Proof of Lemma 7.10. The proof of Lemma 7.10 follows exactly the lines of the proof of [44, Lemma 5]. For

the reason of completeness we give here the main ideas. Let X± ∈ UV±:= X± ∈ V± | ‖X±‖± ≤ cX for some small

constant cX > 0. We show the assertions of the Lemma only for G+, since the proof of the corresponding statements

for G− proceeds in the same way. According to its definition, G+ maps on a continuous function, if X+ is continuous.

We calculate

f(

q+ +X+

)

− f(

q+)

−Df(

q+)

·X+

=

Ac(X+ + q+)[

Xuu+ (Xss

+ + qss+ )]

−Ac(q+)Xuu+ qss+

λAe(Xc+ + qc+, X

e+ + qe+)[X

e+ + qe+]

2 − λAe(qc+, qe+)(q

e+)

2

−λDcAe(qc+, q

e+)(q

e+)

2Xc+ − λ

[

DeAe(qc+, q

e+)(q

e+)

2 + 2Ae(qc+, qe+)q

e+

]

Xe+

+λ[

B(X+ + q+)[

(Xss+ + qss+ )

]

−B(q+)qss+

]

Xuu+

Ass(X+ + q+)[

Xss+ + qss+

]

−Ass(q+)[

Xss+ + qss+

]

−DAss(q+)X+qss+

Auu(X+ + q+)Xuu+ −A

uu(q+)Xuu+

. (7.43)

Exemplarily for the uu-component we deduce

Auu(X+ + q+)Xuu+ −A

uu(q+)Xuu+ = [Auu(Xc

+ + qc+, Xe+, X

ss+ + qss+ , X

uu+ )−Auu(q+)]X

uu+

=

∫ 1

0

DAuu(σX+ + q+) dσX+Xuu+

= O(

|X+| |Xuu+ |)

.

Here we exploited that ‖X±‖± ≤ cX and the boundedness of q+. Similarly we obtain the remaining estimates. These

estimates show that G± map X ∈ V± onto an (Xc, λ Xe, Xss, Xuu) with (Xc, Xe, Xss, Xuu) ∈ V±.

Next we prove the differentiability of G+ with respect to X+ for all X+ ∈ UV+ . We show that DXG+(X+)(·) is

given by the linear operator Df(

q+ +X+

)

(·) −Df(

q+)

(·). Indeed, the following computations show that Df(

q+ +

X+

)

(·)−Df(

q+)

(·) is an element of L(V+). Let Ze ∈ V e+ and X+ ∈ V+ with ‖X+‖+ ≤ cX . We consider exemplarily

Def(

q+ +X+

)

Ze −Def(

q+)

Ze. Exploiting (7.43) it follows that

‖Def(

q+ +X+

)

Ze −Def(

q+)

Ze‖e+

=∥

∥

∥λ(

2Ae(Xc+ + qc+, X

e+ + qe+)[X

e+ + qe+] +DeA

e(Xc+ + qc+, X

e+ + qe+)[X

e+ + qe+]

2

−DeAe(qc+, q

e+)(q

e+)

2 − 2Ae(qc+, qe+)q

e+ +DeB(X+ + q+)

[

(Xss+ + qss+ )

]

Xuu+

)

Ze∥

∥

∥

e

+


= supt∈[0,L]

e−αet∣

∣λ(

2Ae(Xc+ + qc+, X

e+ + qe+)[X

e+ + qe+]

+DeAe(Xc

+ + qc+, Xe+ + qe+)[X

e+ + qe+]

2 −DeAe(qc+, q

e+)(q

e+)

2

− 2Ae(qc+, qe+)q

e+ +DeB(X+ + q+)

[

(Xss+ + qss+ )

]

Xuu+

)

Ze∣

∣

≤ |λ| · c · supt∈[0,L]

e−αet∣

∣

∣ eαet‖Xe

+ + qe+‖e+ + e2α

et(‖Xe+ + qe+‖

e+)

2 + eαet‖qe+‖

e+

+ e2αet(‖qe+‖

e+)

2 + eαsst‖Xss

+ + qss+ ‖ss+ · e

αuu(t−L)‖Xuu+ ‖

uu+

∣

∣

∣‖Ze‖e+

≤ |λ| · c cX · ‖Ze‖e+,

for some constant c, that is independent of L. The other components are treated similarly. Finally a straightforward

calculation shows that

1

‖Z‖+

∥

∥

∥G+(X+ + Z)−G+(X+)−Df(

q+ +X+

)

(Z)−Df(

q+)

(Z)∥

∥

∥

+→ 0,

whenever ‖Z‖+ → 0, Z ∈ V+. This shows that Df(

q+ +X+

)

(·)−Df(

q+)

(·) is indeed the derivative of G+ at X+.

The differentiability w.r.t. z = ϕ, ξe, µ, λ can be shown analogously. For higher derivatives the procedure is similar.

Next we solve Equation (7.42) with d(L) := (0, −qe+(L), −qss+ (L), quu− (−L)). Analogously to [44, Proposition 4] we

find:

Lemma 7.11. Let L > L0 be fixed. Then there exists a unique solution, X = (X+,X−) ∈ V , of Equation (7.42),

G±(d(L), ϕ, ξe, δ, µ, λ,X+, X−) = 0.

This solution satisfies the coupling condition (7.5),

X+(L)− X−(−L) =(

0, −qe+(L), −qss+ (L), quu− (−L)

)

.

Furthermore X depends smoothly on z = L,ϕ, ξe, µ, λ. Moreover there exists a constant cX such that for any in terms

of the absolute value small ϕ, ξe, δ, µ, λ and any L > L0 the operator norm of DzX± is bounded by cX :

‖DzX±(L,ϕ, ξe, δ, µ, λ)(·)‖± ≤ cX . (7.44)

For the sake of brevity we introduce the following notation X(L,ϕ, ξe, δ, µ, λ)(·) = X(L).

Proof of Lemma 7.11. The proof of Lemma 7.11 proceeds similarly to the proof of [44, Proposition 4]. Let L > L0

be fixed and sufficiently large. According to (7.41), solutions of (7.42) correspond to fixed points of the operator:

NL(d(L), X+, X−) := NL(d(L), G+(X+), G−(X−)).

At first we check the prerequisites of the Banach Fixed Point Theorem. According to Lemma 7.9 there is a CN such

that∥

∥NL(d(L), X+, X−)∥

∥

V≤ CN

[

‖d(L)‖+ ‖G±(X±)‖V]

.

7.5 The fixed point equation 125

According to Lemma 7.10, there is a constant CG such that ‖G±(X±)‖V ≤ CG ·(

‖X±‖V + ‖q±‖V)

‖X±‖V . If

‖X±‖± < cX , then by choosing |ξe|, δ, cX and Jλ sufficiently small the operator NL maps the set UV±= X± ∈

V± | ‖X±‖± ≤ cX into itself.

Further, according to the definition of G± the derivative DX±G±(0) = 0. Consequently D(X+,X−)NL(0, 0, 0) = 0.

Hence NL is contractive on UV±, if cX is sufficiently small. Thus the Banach Fixed Point Theorem yields a fixed point

X(L) of (7.42).

To show the smooth dependence on the parameters, we apply the Implicit Function Theorem to (7.42). The same

arguments as above yield

D(X+,X−)G±(0, ϕ, ξe, δ, µ, λ, 0, 0) = id,

where id is the identity. Recall that d(L) → 0, if L tends to ∞. Consequently we may apply the Implicit Function

Theorem for Banach spaces, (see e.g. [12]), for any fixed, but sufficiently large L > L0. This yields a unique, Ck-smooth

solution, X(L) ∈ UX ⊂ V , of (7.42) within a neighbourhood UX = X± ∈ V± | ‖X±‖± ≤ cX of zero.

Note that C0(I±,R) depend on L. However, Lemma 7.9 states that the norm of NL and its derivatives are uniformly

bounded in L. Thus the neighbourhoods UX can be chosen uniformly in L. Hence for each L we may repeat the above

arguments. This implies that X(L) exists for all L ∈ (L0,∞), if L0 is chosen sufficiently large. Further we note that

L0 may be chosen independent of λ.

Next we consider the differentiability with respect to λ. Since the spaces V e± depend on λ, the smoothness in λ follows

not immediately by the Implicit Function Theorem. Recall that the spaces V± are subsets of C0(I±,R). Hence we

may solve the fixed point equation (7.42) not in V±, but also in C0(I±,R). (In fact the only reason to consider

(7.42) in V±, is that this instantly yields that X satisfies the exponential rates of the spaces V±.) Since C0(I±,R) is

independent of λ, the Implicit Function Theorem yields a solution that depends smoothly on λ. But V± are subsets

of C0(I±,R) and thus the uniqueness property of this solution implies that it has to coincide with X.

It remains to investigate the smoothness of X with respect to L. Since the spaces V± depend on L, we use a time

rescaling argument, as it is done in [44]. Let ε ∈ R be small and define t := t1+ε . Applying this transformation to

(2.8) we obtain the rescaled system

v( t ) = (1 + ε)f(v( t )).

Next we repeat all the steps done so far and define analogously to the operator GL an operator GεL and consider the

fixed point equation

GεL(d(L), ϕ, ξe, µ, λ,Xε

+, Xε−) = 0.

Denote by Xε(L) the solution to the rescaled fixed point equation, where L > L0. Since both, X(L) and X

ε(L) are

unique solutions, it follows that

Xε(L) = X((1 + ε)L).

We infer from the above line that

DεXε(L) =

d

dεXε(L) =

d

dεX((1 + ε)L) = DLX((1 + ε)L) · L,

supposed that DεXε exists. However, the vector field (1 + ε)f(·) depends smoothly on ε. Hence the differentiability

of Xε can be shown in the same way as we proved the differentiability with respect to λ.

Finally we give an argument for (7.44), that is the operator norm of DzX is uniformly bounded in L. Note that for

fixed L,ϕ, ξe, δ, µ, λ the mapping DzX±(L,ϕ, ξe, δ, µ, λ) ∈ L(R, V ), z = ϕ, ξe, δ, µ, λ. Since R is finite dimensional, it


is immediate that the (operator) norm of DzX±(L,ϕ, ξe, δ, µ, λ) is bounded. Thus it remains to show the uniformity.

To this end we differentiate exemplarily (X+,X−) = NL(d,X+,X−) with respect to λ:

(DλX+, DλX−) = DλNL(d,G+(X+), G−(X−)) +DdNL(d,G+(X+), G−(X−)) ·( d

dλd,

d

dλd)

+DgNL(d,G+(X+), G−(X−)) ·[

(DλG+(X+), DλG−(X−))

+ (DX+G+(X+) ·DλX+, DX−

G−(X−) ·DλX−)]

.

Recall that if X ∈ UX , that is ‖X‖V ≤ cX . According to Lemma 7.9 the norms ‖DλNL‖V , ‖DdNL‖V and ‖DgNL‖L(V )

are uniformly bounded in L. Lemma 7.4 shows that ddλd =

(

0,O(eαeL),O(eα

ssL),O(e−αuuL)

)

and therefore ‖DdNL ·(

ddλd,

ddλd)

‖V is uniformly bounded. Combining this with the estimates of Lemma 7.10 and exploiting ‖X‖V ≤ cX

gives the assertion. The other derivatives can be treated similarly.

7.6 The estimates of the solutions X±

In what follows let L > L0 be fixed. To prove Theorem 2.4 it remains to show that the solutions v± = q±+X± satisfy

the claimed exponential rates, where X± is given in Lemma 7.11. Lemma 7.4 yields the corresponding rates for q±.

Hence it remains to show that also X satisfies those exponential rates. Define

X± := (Xc±,Xc±,X

ss± ,X

uu± ).

Lemma 7.12. If X = (X+,X−) is the unique solution of Lemma 7.11, the following estimates are satisfied:

(i) ‖Xuu+ ‖uu+ = O

(

e−αuuL)

and ‖Xss− ‖ss− = O

(

eαssL)

,

(ii) ‖Xe+‖e+ = λO

(

e−αuuL)

and ‖Xe−‖e− = λO

(

eαeL)

,

(iii) ‖X+‖∞ = O(

e−αL)

and ‖X−‖∞ = λO(

eαeL)

+O(

e−αL)

.

Proof of Lemma 7.12 At first we observe that (i) can be obtained totally analogously to [44, Lemma 6]. The

proof of (ii) follows also the lines of the proof of [44, Lemma 6]. We exploit that X is a fixed point of the operator

NL((0, −qe+(L), −q

ss+ (L), quu− (−L)), ·, ·). Further we use Lemma 7.5, (7.27), (7.35) to get that for t ∈ I+:

Xe+(t) = P

e[

Φ+(t, L)Pc+(L)(a

c, 0, 0, 0) + Φ+(t, L)Puu+ (L)(0, 0, 0, auu)

+ Pe∫ t

L

Φ+(t, σ)[

P c+(σ) + Puu+ (σ)]

G+(X+)(σ) dσ

+ Pe∫ t

0

Φ+(t, σ)[

P e+(σ) + P ss+ (σ)]

G+(X+)(σ) dσ]

= Φe+(t, L)[

λO(

eαeL)

ac]

+ λ

∫ t

L

Φe+(t, σ)DcAe(qc+, q

e+)(q

e+(σ))

2ac dσ + Φe+(t, L)

[

λO(

e−αL)

auu]

+ λ

∫ t

L

Φe+(t, σ)DcAe(qc+, q

e+)(q

e+(σ))

2[

O(

e−αL)

auu +

∫ σ

L

DcAc(q+)q

ss+ (ω)Φuu+ (ω,L)auu dω

]

dσ

+ λ

∫ t

L

Φe+(t, σ)B(q+(σ))qss+ (σ)Φuu+ (σ, L)auu dσ

7.6 The estimates of the solutions X± 127

+ Pe∫ t

L

Φ+(t, σ)(

Gc+ +O(

e−ασ)

Guu+ , λ2O(

eαeσ)

Gc+ + λO(

e−ασeαeσbig)Guu+ ,O

(

eαssσ)(

Gc+ +Guu+)

, Guu+

)

dσ

+ Pe∫ t

0

Φ+(t, σ)(

0, Ge+ + λ2O(

eαeσ)

Gc+ + λO(

e−ασ)

Guu+ , Gss+ +O(

eαssσ)(

Gc+ +O(

e−αeσ)

Ge+ +Guu+)

, 0)

dσ.

Next we apply Lemma 7.4, Lemma 7.10 and (i), (7.32), (7.34) and the definition of the space V+ to estimate the above

line further

Xe+(t) = λO

(

eαet)

O(

e−αL)

auu + λ

∫ t

L

O(

eαe(t+σ)

)

O(

e−αL)

auu dσ + λO(

e−L)

auu

+ λ

∫ t

L

O(

eαe(t+σ)

)

O(

e−αL)

auu dσ + λ

∫ t

L

O(

eαe(t−σ)

)

O(

eαssσ)

O(

eαuu(σ−L)

)

auu dσ

+ λ2∫ t

L

O(

eαe(t−σ)

)

O(

eαeσ)

eαuu(σ−L)‖Xuu+ ‖

uu+ eα

ssσ‖Xss+ + qss+ ‖ss+ dσ

+ λ

∫ t

L

O(

eαe(t−σ)

)

O(

e−ασeαeσ)


uu+ ‖X+‖+ dσ

+ λ2∫ t

L

∫ t

σ

O(

eαe(t+ω)

)

[


uu+ eα

ssσ‖Xss+ + qss+ ‖ss+

+O(

e−ασ)


uu+ ‖X+‖+

]

dω dσ

+ λ2∫ t

L

∫ t

σ

O(

eαe(t+ω)

)

∫ ω

σ

O(

eαss)

O(

eαuu(−σ)

)


uu+ ‖X+‖+ d dω dσ

+ λ

∫ t

L

O(

eαe(t−σ)

)

∫ t

σ

O(

eαssω)

O(

eαuu(ω−σ)

)


uu+ ‖X+‖+ dω dσ

+ λ

∫ t

0

O(

eαe(t−σ)

)

[

O(

e2αeσ)

‖Xe+ + qe+‖e+ ‖X

e+‖

e+

+ λ eαuu(σ−L)‖Xuu+ ‖

uu+ eα

ssσ‖Xss+ + qss+ ‖ss+ +O

(

e−ασ)


uu+ ‖X+‖+

]

dσ.

Finally this yields

Xe+(t) = λO

(

e−(αuu+)L)

+O(

eαet)

‖Xe+ + qe+‖e+ ‖X

e+‖

e+.

Consequently

‖Xe+‖e+ = λO

(

e−αuuL)

+ · supt∈[0,L]

O(

e(αe−αe)t

)

‖Xe+ + qe+‖e+ ‖X

e+‖

e+.

The term supt∈[0,L]

O(

e(αe−αe)t

)

is uniformly bounded in L ∈ [L0,∞) for all λ ∈ Jλ. Exploiting again that

‖X+‖+ ≤ cX , we can choose cX , δ such that ‖Xe+ + qe+‖e+ ≤ cX + δ < 1. Hence

‖Xe+‖e+ = λO

(

e−αuuL)

.

Similarly, for t ∈ I−, we estimate

Xe−(t) = P

e[

Φ−(t,−L)Pe−(−L)(0, a

e, 0, 0) + Φ−(t,−L)Pss− (−L)(0, 0, ass, 0)

+ Pe∫ t

−L

Φ−(t, σ)[

P e−(σ) + P ss− (σ)]

G−(X+)(σ) dσ]


= eαe(t+L)ae + λ eα

e(t+L)O(

e−(αe+α)L)

ass +

∫ t

−L

eαe(t−σ)quu− (σ)Φss− (σ,−L)assd σ

+ λ

∫ t

−L

eαe(t−σ)

[

O(

|Xe−|2 + |Xss− | |X

uu− + quu− |

)

+O(

e(αe+α)σ

)

O(

|X−| |Xss− |)]

dσ

+ λ

∫ t

−L

∫ t

σ

eαe(t−ω)quu− (ω)Φss− (ω, σ)O

(

|X−| |Xss− |)

dωdσ

= O(

eαe(t+L)

)

ae + λO(

eαet)

O(

e−αL)

ass +O(

eαe(t+L)

)[

(‖Xe−‖e−)

2 + λ eαssL]

.

By (7.40) we get

ass = O(

eαssL)

and ae = λO(

eαeL)

,

since dss = qss+ (L) and de = qe+(L) and by exploiting Lemma 7.10 and (i). As for X+ this yields the assertion.

The proof of (iii) proceeds as the proof of corresponding assertions in [44, Lemma 6]. Observe that the term λO(

eαeL)

in ‖X−‖∞ is due to the estimate of Xe− and ae.

To finish the proof of Theorem 2.4, it remains to estimate Xk±(0), k = c, ss, uu. Basically this can be done as in [44].

To highlight the effect of the e-component on this estimates, we do this exemplarily for Xuu+ (0) and Xe−(0). We exploit

that Xuu+ (0) is defined by (7.35). Using the from of the projections given in Lemma 7.5 yields

Xuu+ (0) = PuuΦ+(0, L)P

c+(L)(a

c, 0, 0, 0) + PuuΦ+(0, L)Puu+ (L)(0, 0, 0, auu)

+ Puu∫ 0

L

Φ+(0, σ)[

P c+(σ) + Puu+ (σ)]

G+(X+, ϕ, ξe)(σ) dσ

= Φuu+ (0, L)auu +

∫ 0

L

Φuu+ (0, σ)Guu+ (X+, ϕ, ξe)(σ) dσ.

We exploit Lemma 7.10 to estimate G+(X+)(σ). To estimates the norms of X+ we apply Lemma 7.12. We get

Xuu+ (0) = Φuu+ (0, L)

(

duu +O(

e−αuuL)

‖d‖∞ +O(

e−αuuL)

‖(

G+(X+), G−(X−))

‖V)

+

∫ 0

L

Φuu+ (0, σ)O(

|Xuu+ (σ)| |X+(σ)|)

dσ

= Φuu+ (0, L)(

quu− (0) +O(

e−αuuL)[

λO(

eαeL)

+O(

e−αL)]

+O(

e−αuuL)

O(

‖X+‖∞ + ‖X−‖∞)

)

+

∫ 0

L

e−αuuσeα

uu(σ−L)‖Xuu+ ‖uu+ O

(

‖X+‖∞)

dσ

= Φuu+ (0, L)(

[

∆uue−αuuL +O

(

e−2αuuL)]

+O(

e−αuuL)[

λO(

eαeL)

+O(

e−αL)]

)

+O(

e−αuuL)

∫ 0

L

O(

e−αuuL)

O(

e−αL)

dσ.

Finally we recall the estimate of the transition matrix Φuu+ , (7.32). Moreover, we observe that Lemma 7.1 shows that

there is a smooth, bounded, nonzero function ∆uu = ∆uu(L,ϕ, ξe, δ, µ, λ) such that Φuu+ (0, L)∆uu = ∆uu e−αuuL for

all t ≥ 0. Hence

Xuu+ (0) = Φuu+ (0, L)

[

∆uue−αuuL + λO

(

e(αuu−αe)L

)

+O(

e(αuu+α)L

)]

= ∆uu e−2αuuL + λO(

e−(2αuu−αe)L)

+O(

e−(2αuu+α)L)

.

7.6 The estimates of the solutions X± 129

Observe that 2αuu < 2αuu − αe, whenever we choose ν < |αe|/2αuu and λ 6= 0. Consequently ∆uue−2αuuL denotes

the leading order term of Xuu+ (0) for all λ ∈ Jλ. In the same way we estimate

Xe−(0) = P

e[

Φ−(0,−L)Pc−(−L)(a

c, 0, 0, 0) + Φ−(0,−L)Pe−(−L)(0, a

e, 0, 0) + Φ−(0,−L)Pss− (−L)(0, 0, ass, 0)

]

+ Pe∫ 0

−L

Φ−(0, σ)[

P c−(σ) + P e−(σ) + P ss− (σ)]

G−(X−, ϕ, ξe)(σ) dσ

= Φe(0,−L)ae +

∫ 0

−L

Φe(0, σ)[

Ge−(X−, ϕ, ξe)(σ) + λO

(

e(αe+α)σ

)

Gss− (X−, ϕ, ξe)(σ)

+ λ

∫ 0

σ

e−αeωquu− (ω)Φss(ω, σ)Gss− (X−, ϕ, ξ

e)(σ)dω]

dσ

= eαeL[

qe+(L) + λO(

e−L)

‖d‖∞ + λO(

eαeL)

‖Ge+‖e+ + λO

(

eαeL)

‖Gc+‖c+ + λO

(

e−αL)

‖(G+, G−)‖V]

+ λ

∫ 0

−L

e−αeσe2α

e(σ+L)(‖Xe−‖e−)

2 +O(

eαss(σ+L)eα

uuσ)

‖Xss− ‖ss− ‖X

uu− + quu− ‖

uu−

+O(

e(αe+α)σ

)

eαss(σ+L)‖Xss− ‖

ss− ‖X−‖−d σ

+ λ

∫ 0

−L

∫ 0

σ

e−αeωO

(

eαuuω)

eαss(ω−σ)eα

ss(σ+L)‖Xss− ‖ss− ‖X−‖−dωdσ

= eαeL[

qe+(L)]

+ λO(

e−L)

+ λ

∫ 0

−L

e−αeσe2α

e(σ+L)λ2O(

e2αeL)

d σ

= λ ξe∆ee2αeL + λ2O

(

e(αe+2αe)L

)

+ λO(

e−L)

.

Similarly we obtain Xc+(0),X

c−(0),X

ss− (0) so that

Xc+(0) = O

(

e−(αuu+α)L)

,

Xuu+ (0) = ∆uue−2αuuL + λO

(

e−(2αuu−αe)L)

+O(

e−(2αuu+α)L)

,

Xc−(0) = O

(

e(αss−α)L

)

,

Xe−(0) = λ ξe∆ee2α

eL + λ2O(

e(αe+2αe)L

)

+ λO(

e−L)

,

Xss− (0) = ∆sse2α

ssL + λO(

e(2αss+αe)L

)

+O(

e(2αss−α)L

)

. (7.45)

Moreover, according to Lemma 7.7, Xe+(0) = 0, Xss+ (0) = 0, Xuu− (0) = 0. Finally we recall that the solutions v of

system (2.8) is composed of

v(t) =

v+(t+ L), t ∈ [−L, 0],

v−(t− L), t ∈ [0, L],where

v+(t) := q+(t) + X+(t), t ∈ I+,

v−(t) := q−(t) + X−(t), t ∈ I−.

Exploiting Lemma (7.4), the conditions (7.4) and the estimates (7.45), we obtain

v(−L) :=

ϕ− L

λ ξe

δ

∆uue−2αuuL

+

O(

e−(αuu+α)L)

0

0

λO(

e−(2αuu−αe)L)

+O(

e−(2αuu+α)L)

,


v(L) :=

L+ ϕ

λ ξe∆ee2αeL

∆sse2αssL

δ

+

O(

e(αss−α)L

)

λ2O(

e(αe+2αe)L

)

+ λO(

e−L)

λO(

e(2αss+αe)L

)

+O(

e(2αss−α)L

)

0

.

The estimates of the derivatives of v(·) can be obtain similarly, where one has to derive corresponding estimates of the

operator G± for higher derivatives.

This finishes the proof of Theorem 2.4.

131

Chapter 8

Discussion and conclusions

In this thesis we have contributed to the analytical theory of Homoclinic Snaking (of 1-homoclinic orbits) in ODE

systems without the particular structure of first integrals or reversible symmetries. Starting point of our analysis is the

work in [3]. There a series of hypotheses (formulated to an ODE system in R4) is presented and proved to be sufficient

for the occurrence of Homoclinic Snaking. The crucial structural assumptions in this framework are the existence of

a heteroclinic EtoP cycle that connects a hyperbolic equilibrium E with a hyperbolic periodic orbit P , the presence

of a first integral, as well as, a reversible structure of the underlying ODE.

In Chapter 3 and Chapter 4 we have considered two perturbation scenarios of this setup:

At first we have investigated perturbations that break the reversible structure, but keep the first integral in Chapter 3.

We have shown that such perturbations may produce two distinguished bifurcation diagrams of homoclinic orbits:

isolas or criss-cross snaking. A natural question is to consider the transition between these both bifurcation diagrams.

We have briefly discussed this topic, but gave no rigorous analysis.

In Chapter 4 we have considered a perturbation scenario, where the reversible structure, as well as, the first integral

disappear. We have shown that, as in Chapter 3, the resulting bifurcation diagram consists of isolas or criss-cross

snaking. But this time the bifurcation diagram involves not only the snaking parameter, but also the perturbation

parameter. This is due to the loss of structure in the perturbed ODE, which effects that the continuation of homoclinic

orbits takes generically place in 2 parameters.

In Chapter 5 we have considered the behaviour of homoclinic orbits near EtoP cycles in an ODE system that has

from the beginning no particular structure (in contrast to the underlying systems in Chapter 3 and Chapter 4). We

have analysed how the behaviour of the primary EtoP cycle determines the continuation behaviour of the homoclinic

orbits. In our considerations we have distinguished the cases that the periodic orbit P has positive or negative

Floquet multipliers, respectively. For positive Floquet multipliers we have discussed two different scenarios. First, in

Section 5.1 and 5.2, we have verified Homoclinic Snaking as it was previously observed numerically in the motivating

example (1.6). Furthermore we have described a nonsnaking scenario consisting of disjoint isolas in Section 5.4. To

our knowledge such a behaviour has not yet been observed in systems in R3. In systems in R4 however this effect was

observed numerically [7, 50]. For negative Floquet multipliers we have studied the corresponding snaking scenario in

Section 5.3.

To perform our analysis in Chapter 3 - Chapter 5 we have formulated a series of hypotheses that have been observed

in numerical experiments, such as reversible symmetries or the existence of first integrals. Further have we assumed

that the considered vector field possesses certain geometric properties, regarding the intersection of the manifolds of

the equilibrium E and the periodic orbit P . Those geometric assumptions have yet not been verified numerically, with

the exception of the assumptions in Chapter 5, which have been numerically proved for the example (1.6) by Rieß

in [37, Section 6]. A remaining task is to validate those assumptions numerically in further model equations.

132 8 Discussion and conclusions

Beside that we note that our analysis in Chapter 3 - Chapter 5 is restricted to R4 and R3, respectively. If one is

interested in similar problems in higher dimensional state spaces, then one has to take into account that the number

of bifurcation equations to solve increase accordingly. This has become apparent in Chapter 4, where the number of

bifurcation equations increased by two, due to the loss of the first integral. To handle such situations would be a field

for future research. A promising approach seems to be Lin’s method combined with the Fenichel coordinates. A first

step in this respect yields [57], where Lin’s method is adapted to continue heteroclinic orbits that involve periodic

orbits. There the coupling near the periodic orbit is done by means of a Poincare map instead of exploiting Fenichel

coordinates. Similarly investigations have been done in [54], where Lin’s method has been used to study homoclinic

solutions near an EtoP cycle. There, in contrast to [57], a variational approach has been exploited to realise the

coupling near the periodic orbit.

A further interesting question is the detection of N-homoclinic orbits to the equilibrium E, these are orbits that follow

the primary EtoP cycle N times before returning to E. Then also couplings near E have to be considered. For the

case of symmetric 2-homoclinic orbits we refer to the work of Knobloch et al. in [35], where it is shown that within

the setup of [3], symmetric 2-homoclinic orbits correspond to isolas in the bifurcation diagram. However, the analysis

there relies on the reversible structure of the underlying ODE and is (so far) restricted to 2-homoclinic orbits in R4.

If one is interested in N-homoclinic orbits in higher dimensional systems, then again an approach that combines Lin’s

method with the Fenichel coordinates seems to be appropriate. Finally we refer to the numerical results in [58], which

indicates that N-homoclinic orbits exist also in the example (1.6), which possesses no particular structure.

To formulate the hypothesis of our setup we have exploited Fenichel coordinates near a periodic orbit. The existence

of those coordinates near hyperbolic periodic orbits can be derived from the existing literature. In Chapter 6 Fenichel

coordinates have been constructed near a 1-parameter family of periodic orbits, in which the dimensions of the

corresponding stable manifolds change when changing the family parameter. To this end a foliation of an extended

(stable) manifold has been derived. Furthermore we have shown that this foliation is smooth even in the limit when

the weak (stable) Floquet exponent of the periodic orbit tends to zero.

To carry out our continuation analysis of homoclinic orbits, we have solved a Shilnikov problem near the 1-parameter

family of periodic orbits in Chapter 4. The analysis of this Chapter is based on [44], where a Shilnikov problem in

the context of slow/fast systems has been solved. We adapted this approach to the situation of periodic orbits with a

weak (stable) Floquet exponent that tends to zero.

To address again the extension of Homoclinic Snaking in higher dimensional spaces, we note that both corresponding

Fenichel coordinates and a corresponding solution of the Shilnikov problem have to be derived. The construction of

Fenichel coordinates in higher dimensional spaces should proceed similar to the construction in Chapter 6. Regarding

the Shilnikov problem in higher dimensions, it should be noticed that it suffices in general to derive estimates with

rates that merely corresponds to the leading stable, centre and unstable Floquet multipliers of P . This in fact should

work rather similar to the constructions in Chapter 4.

133

Acknowledgements

I wish to thank my supervisor Jurgen Knobloch for his advice and support during the last four years, his inexhaustible

efforts to help me finishing this work and, of course, for the many re(a)d pages. Moreover I like to thank the German

National Academic Foundation (Studienstiftung des deutschen Volkes) for the support in the last years.

Zum Schluss bleibt nur noch Danke zu sagen. Danke an alle, die mir geholfen haben, mich auf meinen mathematischen

und insbesondere all den anderen kleinen und großeren Umwegen beim Anfertigen dieser Arbeit nicht zu verirren. Und

so ist auch eine Erkenntnis dieser Arbeit:

Deine menschliche Umgebung ist es, die das Klima bestimmt.

Mark Twain

134 8 Discussion and conclusions

BIBLIOGRAPHY 135

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Erklarung

Ich versichere, dass ich die vorliegende Arbeit ohne unzulassige Hilfe Dritter und ohne Benutzung anderer als der

angegebenen Hilfsmittel angefertigt habe. Die aus anderen Quellen direkt oder indirekt ubernommenen Daten und

Konzepte sind unter Angabe der Quelle gekennzeichnet.

Bei der Auswahl und Auswertung folgenden Materials haben mir die nachstehend aufgefuhrten Personen in der jeweils

beschriebenen Weise (entgeltlich/unentgeltlich) geholfen:

PD Dr. Jurgen Knobloch (Betreuer, unentgeltlich)

Weitere Personen waren an der inhaltlich-materiellen Erstellung der vorliegenden Arbeit nicht beteiligt. Insbesondere

habe ich hierfur nicht die entgeltliche Hilfe von Vermittlungs- bzw. Beratungsdiensten (Promotionsberater oder an-

derer Personen) in Anspruch genommen. Niemand hat von mir unmittelbar oder mittelbar geldwerte Leistungen fur

Arbeiten erhalten, die im Zusammenhang mit dem Inhalte der vorgelegten Dissertation stehen.

Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ahnlicher Form einer Prufungsbehorde vorgelegt.

Ich bin darauf hingewiesen worden, dass die Unrichtigkeit der vorstehenden Erklarung als Tauschungsversuch bewertet

wird und gemaß Paragraph 7 Abs. 8 der Promotionsordnung den Abbruch des Promotionsverfahrens zur Folge hat.

Ilmenau, den 11.07.2014 .........................................

nonreversible homoclinic snaking scenarios€¦ · solutions that exist in a neighbourhood of what...

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