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https://doi.org/10.1088/0951-7715/29/11/3346

http://iopscience.iop.org/article/10.1088/1361-6544/aa7e9b

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3346

Nonlinearity

Symbolic coding for noninvertible systems: uniform approximation and numerical computation

Wolf-Jürgen Beyn, Thorsten Hüls and Andre Schenke

Department of Mathematics, Bielefeld University, POB 100131, 33501 Bielefeld, Germany

E-mail: [email protected], [email protected] and [email protected]

Received 29 July 2015, revised 5 April 2016Accepted for publication 9 May 2016Published 8 September 2016

Recommended by Dr Dmitry Turaev

AbstractIt is well known that the homoclinic theorem, which conjugates a map near a transversal homoclinic orbit to a Bernoulli subshift, extends from invertible to specific noninvertible dynamical systems. In this paper, we provide a unifying approach that combines such a result with a fully discrete analog of the conjugacy for finite but sufficiently long orbit segments. The underlying idea is to solve appropriate discrete boundary value problems in both cases, and to use the theory of exponential dichotomies to control the errors. This leads to a numerical approach that allows us to compute the conjugacy to any prescribed accuracy. The method is demonstrated for several examples where invertibility of the map fails in different ways.

Keywords: noninvertible dynamical systems, homoclinic orbits, symbolic coding, numerical computationMathematics Subject Classification numbers: 65P20, 37B10, 65Q10, 37C29

(Some figures may appear in colour only in the online journal)

1. Introduction

Quite a few fundamental theorems on invertible dynamical systems have found suitable ana-logs in the theory of noninvertible systems. In this paper we reconsider corresponding gener-alizations of the well-known homoclinic theorem (see Smale [40] and Šil’nikov [39]) on the symbolic coding of maximal invariant sets near a transversal homoclinic point. Our main goal

W-J Beyn et al

Symbolic coding for noninvertible systems: uniform approximation and numerical computation

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is to set up and analyze a numerical approach that allows us to compute all of the symbolically coded orbits near a homoclinic orbit of a noninvertible system up to a prescribed tolerance.

The motivation for studying noninvertible maps comes from different sources. One major source is maps in infinite-dimensional Banach spaces that arise as flow or Poincaré maps of partial differential equations and delay differential equations. We mention the work of Hale and Lin [12] and, in particular, the ensuing work of Steinlein and Walther [41, 42], which gives a fairly complete picture of shadowing and symbolic coding near homoclinics in the nonin-vertible case. In the work by Kalkbrenner [22] (see also the survey in [2]), the closely related notion of exponential dichotomy was generalized to noninvertible and nonautonomous systems and then applied to characterize symbolic coding near homoclinics. The proposed notion of a ‘regular exponential dichotomy’ agrees with the earlier definition by Henry [14, chapter 7], and the proof of symbolic coding employs Palmer’s method for proving the homoclinic theorem for invertible systems [31, 32]. We also mention modern results on so-called generalized Hénon maps from [9, 11], as well as the work of Sander [35], which generalizes the notions of hyper-bolicity and homoclinic tangles from nonlinear maps to nonlinear relations.

A second source of noninvertibility is reduced models of higher-dimensional dynamical systems, which allow us to study chaotic dynamics in low dimensions. A characteristic exam-ple, which will also be treated in this paper, is a noninvertible two-dimensional model for ‘wild chaos’, derived in [5] from a 5-dimensional invertible system and investigated exten-sively in [15, 16].

In the following, we describe the contents of this paper in more detail by putting some emphasis on our computational approach. In particular, we treat finite orbit segments that approximate the symbolic coding of noninvertible maps and provide an alternative road for proving the homoclinic theorem in the biinfinite case.

Since we aim at numerical computations, we consider a finite-dimensional smooth map f : d d→R R and its induced dynamical system

x f x x n, , .n n nd

1 ( ) R Z= ∈ ∈+

Moreover, we assume that we are given a fixed point

f d( ) Rξ ξ= ∈

and an associated nontrivial homoclinic orbit x xn n¯ ( ¯ )Z Z= ∈ , i.e.

x f x n x x n, , lim , for some .n nn

n n1¯ ( ¯ ) ¯ ¯ →

Z Zξ ξ= ∈ = ≠ ∈+±∞

Our basic assumption states that the variational equation

u Df x u n,n n n1 ( ¯ ) Z= ∈+ (1)

has an exponential dichotomy on Z according to definition 18. We emphasize that this defi-nition does not assume that Df xn( ¯ ) is invertible, but we refrain from using the term ‘regular dichotomy’ from [22] for this situation. Also note that an exponential dichotomy in this sense implies the hyperbolicity of the set

x n: ,n ¯ Zξ= ∪ ∈H (2)

as defined in [42, definition 1.1], and in fact both properties are equivalent in the finite- dimensional case.

In the invertible finite-dimensional case, the condition that (1) has an exponential dichotomy on Z is equivalent to the homoclinic orbit lying on the intersection of the stable and unstable manifolds of the fixed point and this intersection being transversal, see [32, proposition 5.6].

W-J Beyn et alNonlinearity 29 (2016) 3346

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If the intersection is not transversal but tangential, the resulting orbit is called a homoclinic tangency. Homoclinic tangencies show quite intricate dynamical behavior, see [6, 10, 29, 35] and references therein.

For any bounded open set dR⊂O we define the set of orbits remaining in O by

x x x f x nOrb : , for all .dn n n1( ) ( ) ( ) R ZZ

Z= ∈ ∈ = ∈+O O (3)

This set is closed in ZO when endowed with the product topology, and it is invariant under the orbit shift induced by f,

x f x: .n n( ) ( ( ))Z Z= ∈F (4)

It is shown in [42] that there exist a neighborhood O of H, a number N N∈ and a homeomor-

phism h that conjugates : Orb Orb( ) → ( )F O O to a subshift ,A N( )( )Z σΣ :

(5)

Here, S N: 0, , 1N Z Z= − is the space of symbolic sequences on N symbols endowed with the product topology, σ denotes the Bernoulli shift and SA N N( )

Z Z⊂Σ is the topological Markov chain

s S A N j: 1,A N N s s,j j 1 ( ) ( ) ZZ ZΣ = ∈ = ∈+

defined by the matrix

A N

1 1 0 00 0 1 0

00 0 11 0 0

0, 1 .N N,( )

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

= ∈

As in [42], it is not difficult to retrieve the classical homoclinic theorem for the invertible case from this result by adding another conjugacy to the diagram in (5) that assigns to any orbit x Orb( )Z∈ O a single point such as x0.

In our approach we also consider the finite time case S N0, , 1NJ J = … − , where

J n n n n n n, :[ ] ⩽ ⩽ Z= = ∈− + − + , n n−∞ < < < ∞− + . Setting J n n, 1˜ [ ]= −− + , we then

define A nJ

( )Σ by

s S s s A N j J: 0, 1, .A NJ

NJ

n n s s,j j 1 ( ) ˜( )Σ = ∈ = = = ∈− + + (6)

Finite orbits are determined as solutions of boundary value problems

x f x n n n b x x, , , 1, , 0,n n n n1 ( ) ( )= = … − =+ − + − + (7)

with the boundary operators b : d d d→R R R× satisfying a nondegeneracy condition, see sec-tion 4. With the set of finite orbits defined by


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J x x xOrb , : : solves 7 ,JJ

J n n J( ) ( ) ( )= ∈ = ∈O O

our finite time analog of diagram (5) takes the form

(8)

In this diagram we use the shifted interval J n n1 1, 1[ ]− = − −− + and the finite Bernoulli shift

a b b a n J

: ,

, with , for 1.J A N

JA NJ

J J n n

1

1 1

→

( ) ( )σ Σ Σ= ∈ −

−

− + (9)

Note that due to the shift of the interval from J to J − 1, the endpoint condition in (6) is auto-matically satisfied. Furthermore, the map JF in (8) is defined by

x y yx n n

f x n J

: ,

, with, for 1,

, for .

JJ J

J J nn

n

1

1

→

( ) ˜⎧⎨⎩

== −∈

−

−−−

F O O

(10)

This operator satisfies J J: Orb , Orb 1,J ( ) → ( )−F O O , since it preserves the boundary condi-tion. With these settings we obtain a complete finite time analog of the homoclinic theorem in theorem 14.

In the following, we discuss the further results of this paper. In section 2.1, we consider systems with several homoclinic orbits that connect to the same fixed point. We indicate how the finite time conjugacy mentioned above and its numerical computation can be general-ized. In section 2.2, we turn the theoretical approach into a numerical algorithm. For a given

symbolic sequence sJ A NJ

( )∈ Σ , we compute the orbit segment h sJ J( ) in JOrb ,( )O from the solu-tion of an appropriate boundary value problem. Our approximation theory shows that these boundary value problems are well-conditioned in the neighborhood of suitable pseudo-orbits, despite the fact that the map f is noninvertible. This will be demonstrated for a series of exam-ples in section 6.

The theoretical part of our paper is presented in sections 3 and 4. The starting point is the roughness theorem for exponential dichotomies in the noninvertible case (see [22, 14] and theorem 20 in the appendix). We indicate an alternative proof via a direct perturbation argument for the Green function in an exponentially weighted space. The argument follows Sandstede’s idea [36, 37] for flows, which was transferred to invertible mappings in the work of Kleinkauf [23, 25]. Then we consider pseudo-orbits obtained through piecing together the

original homoclinic orbit according to the coding sequence s A N( )ZZ∈ Σ . In theorem 4 we show

that the variational equations from a suitable neighborhood of the pseudo-orbits have an expo-nential dichotomy with data independent of the coding. This result has a complete analog for finite orbit segments, see theorem 6.

With these preparations, we prove the solvability of discrete boundary value problems on both finite and infinite time intervals, see theorem 7. For the infinite case, this result replaces the shadowing lemma for noninvertible maps from [22] (see [30, 34] for invertible maps). We present error estimates for the solutions of finite boundary value problems, which decay towards the interior and are uniform with respect to the coding sequence (corollary 9).


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As a consequence, in theorem 10 we prove the convergence of the solution sets of finite orbits to those of infinite orbits. The convergence holds with respect to the Hausdorff distance

X Y x y x ydist , max sup inf dist , , sup inf dist , ,x X y Y y Y x X

H( ) ( ) ( )⎛

⎝⎜

⎞

⎠⎟=

∈ ∈ ∈ ∈ (11)

where the distance of the finite segments is defined by

x y x ydist , 2 .J Jn n

nn

n n( ) ∑= | − |=

−| |

−

+

(12)

Note that this metric converges as ±∞±n → to the metric

x y x y x ydist , 2 , , ,n

nn n( )Z Z

ZZ Z

Z∑= | − | ∈∈

−| | O

which defines the product topology on ZO . The whole approach extends the earlier approx-imation results for homoclinic tangles in the invertible case [7] and single connecting orbits in the nonautonomous case [19].

In section 5, we show how the homoclinic theorem for the noninvertible case derives natu-rally from these results. A crucial step here is to construct an open neighborhood O such that the orbit set Orb( )O from (3) agrees with the set of orbits coded by symbolic sequences, see theorem 12.

In section 6, numerical computations are shown for the classical (invertible) Hénon exam-ple, a three-dimensional Hénon map, and a modified Hénon example where the dynamics collapses into a line and a snap-back repeller occurs (see [26]). We then use our approach to investigate the chaotic features created by homoclinic orbits in the noninvertible model of ‘wild chaos’ from [5, 15] and a recent model of asset pricing from [8].

2. Numerical computation of the maximal invariant set

We propose an algorithm for the approximation of points from the maximal invariant set

x x: : Orb .c 0 ( )Z= ∈O O (13)

Note that cO is typically a Cantor set on which f-dynamics are semi-conjugate to F -dynamics on Orb( )O , i.e.

where r is surjective. Indeed, if f is invertible, then r turns out to be a homeomorphism, result-

ing in a conjugacy of the f-dynamics on cO to the shift on A N( )ZΣ .

Further note that the stable and unstable manifolds of the fixed point ξ intersect at each point x c∈O transversally, under the hyperbolicity assumptions introduced in section 3. Finding the intersection points of these two manifolds is rather involved. Instead, we calculate the points in cO via homoclinic orbits, which we approximate numerically by solving boundary value problems. The manifolds are only plotted to illustrate the output of this ansatz.

Before the algorithm is introduced in detail, we consider a slightly more general setup.


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2.1. Combining several homoclinic orbits

If only one primary homoclinic orbit is considered, one misses several points of intersection for stable and unstable manifolds, see the left panel in figure 5 for an illustration. Thus, we immediately generalize to multi-humped orbits constructed from different primary homo-clinic orbits xJ

i , i 1, ,= … w.r.t. the same fixed point ξ. It turns out that there exists for each i 1, , ∈ … a number Ni and corresponding neighborhoods that allow the coding of multi-humped orbits with L N1 i i1= + ∑ =

symbols. For details, we refer to theorems 11 and 14. More precisely, we find a homeomorphism between multi-humped orbits in these neighbor-hoods and sequences of L symbols s A N N, ,1( )Z∈ Σ … , where

A N N

I

I

I

, ,

1 1 1 1

1

1

1

1

1

2

( )

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

… =

and Ii denotes the identity in N N1, 1i iR − − , i 1, ,= … . In this section, we do not introduce different notions for finite and infinite time intervals. While the case J Z= is presented here, the finite time case requires suitable boundary conditions, see (6).

We further simplify notions by coding this Markov chain with 1+ symbols s A 1( )Z∈ Σ

A 1

1 1 11

1

0, 1 1, 1( )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

= ∈ + +

and by an extended set

s n s n N, : , 1, , .A A s1 1 0( ) ( ) ( )Z ZΣ = ∈ Σ ∈ …∗

The purpose of the second component n is to serve as a pointer for the current position. This enables us to establish a one-to-one connection between A N N, ,1( )Σ … and

A 1( )Σ∗ .

For s k,A 1

( ) ( )Z ∈ Σ∗ , we define g :

A A N N1 , ,1→( ) ( )Σ Σ∗… :

g s k a s a s a s, : , , , , ,1 0 1( ) ( ( ) ( ) ( ) )Z = …− (14)

where N0 : = 1 and

⎛

⎝⎜⎜

⎞

⎠⎟⎟∑ ∑= … − + ∈ =

= =

−

a s N N i g s k a s: , , 1 , , such that , .ij

s

jj

s

j k0 0

0 0

i i1

Z Z( ) ( ) ( )

Denote by Lσ and σ the Bernoulli shift on A N N, ,1( )Σ … and A 1( )Σ , respectively, and define

s is i N

s i,

, 1 , if ,, 1 , otherwise.

s0( )( ( ) ) ( )Z

Z

Z

⎧⎨⎩

σσ

==

+∗

A direct computation shows that the diagram


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commutes. Consequently, we obtain a conjugacy between this shift-dynamics and the F -dynamics on Orb( )O , where O is a sufficiently small neighborhood of the primary homo-clinic orbits.

2.2. The algorithm

We now introduce our numerical method for computing an approximation of the maximal invariant set cO close to a given point of a homoclinic orbit. Here, we assume that the map f ,d d1( )C R R R∈ × depends on a one-dimensional parameter p and we aim at an approx-imation for the fixed parameter value p. We abbreviate f ( )⋅ for f p,( ¯)⋅ .

The algorithm uses the following steps.

(Ia) Compute a first homoclinic orbit segment xJ1 on a finite interval [ ]= − +J j j, by solving

the boundary value problem (7) using Newton’s method. For a discussion of appropriate boundary conditions, we refer to section 4.1.

(Ib) Calculate a curve of homoclinic f p,( )⋅ -orbits w.r.t. the parameter p by applying numerical continuation techniques, see [1, algorithm 10.2.10]. Detect further homoclinic orbits x x, ,J J

2 … ( N∈ ) lying on this curve at the parameter value p. (II) Choose k Z∈± , k k0< <− + and consider all sequences s k k A, 1[ ] ( )∈ Σ− + satisfying

⩽ ⩽ ⩾ ⩽ ⩽ ⩾ = − = =− +s k i s i k s1, # 1 : 1 1, # 1 : 1 1.i i0 (15)

We refer to the beginning of this section for corresponding notions and to figure 1 for an illustration.

(III) Define the pseudo-orbit w.r.t. the code sequence ( )[ ] ( )+ ∈ Σ−∗

− +s j, 1k k A, 1 by replacing the

symbol 0 with the fixed point ξ and the symbol i with the orbit segment xJi , i 1, ,= … ,

respectively. (IV) Compute a corresponding orbit segment yJ by applying Newton’s method on

J m m: ,ˆ [ ]= − + , m j j k2= − +− − + −, m j j k2= − ++ + − + to the boundary value problem

= = … −=

+ − +

− +

y f y n m m

b y y

, , , 1,

0 , ,n n

m m

1 ( )( )

starting at the pseudo-orbit from step (III). (V) Plot the midpoints y0 of these orbits for all finite code sequences from step (II).

In addition, we compute approximations of the unstable and stable manifolds of the fixed point ξ. For the latter task, we apply a contour algorithm that was introduced in [20] and which applies to a wide class of dynamical systems, including invertible and noninvertible maps.

In section 6, we demonstrate the performance of the algorithm by applying it to several two-dimensional systems and one three-dimensional system, thereby showing various dynam-ical features.

Its applicability is formally justified by the following theorem, in anticipation of assump-tions (A1)–(A4) discussed in section 3 and condition (A5) from section 4.1.


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Theorem 1. Assume that the exact primary homoclinic orbits x x, ,1Z Z… satisfy assumptions

(A1)–(A4), and that the boundary operator b satisfies condition (A5). Then there exist con-stants C, , 0s uβ β > such that for all sufficiently large −j−, j+ the following statement holds true: fix k k0< <− + and let yJ be the solution of the boundary value problem from step (IV ). Then there exists an x c0 ∈O such that

∥ ∥ ⩽ ( )− +β β−− +y x C e e .m m0 0

s u

As a consequence, the approximation errors that occur in step (V) of the algorithm are exponentially small.

In the forthcoming sections, we provide the theoretical background for proving theorem 1 and the homoclinic theorem. For concision, we restrict ourselves to the case of 1= primary homoclinic orbit. After developing this framework, theorem 1 then follows directly by gener-alizing theorem 8 and corollary 9 in a straightforward manner to the case of different primary orbits.

3. Uniform dichotomies of variational equations

In this section, we show that exponential dichotomies along a homoclinic orbit extend in a uniform way to arbitrary orbits obtained by concatenating sufficiently long pieces of the pri-mary orbit. This will enable us to prove a kind of numerical shadowing result in section 3: one finds true orbits in the neighborhood of the concatenated orbits that solve either an infinite or

Figure 1. Upper diagram: a sketch of two homoclinic orbit segments xJ1,2, J = [−2, 2]

and the fixed point ξ, together with their coding in A 4,4( )Σ and A 12( )Σ , respectively. The lower diagram illustrates the construction and coding of multi-humped orbits in the case k 6± =± . The code sequence in the last row—which we introduce formally in section 6.2—represents the number of 0-symbols on the left and right sides of the center hump. The colors of these two numbers symbolize the type of the neighboring hump.


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a finite boundary value problem. Detailed estimates in terms of the metric (12) are given. As in the introduction, our assumptions are as follows.

(A1) f C ,d d1( )R R∈ . (A2) There exist dRξ∈ and x d¯ ( )RZ

Z∈ such that x f xn n1¯ ( ¯ )=+ for all n Z∈ , xk¯ ξ≠ for at least one k Z∈ and xlimn n¯→ ξ=±∞ .

(A3) The point ξ is a hyperbolic fixed point of f, i.e. Df ( )ξ has no eigenvalues on the unit circle (but may have the eigenvalue zero).

(A4) The variational equation

u Df x u n,n n n1 ( ¯ ) Z= ∈+ (16)

has an exponential dichotomy on Z with data K P, ,s u ns u

,,( ¯ ¯ ¯ )α .

Condition (A4) guarantees the hyperbolicity of the set H, introduced in (2) and [42, defini-tion 1.1]. It is not difficult to show that assumptions (A1), (A2) and (A4) imply (A3). However, (A4) does not follow from (A1)–(A3), since it specifies a global property of the set H. Let Qs,u be the projector onto the stable resp. unstable subspace of Df ( )ξ . Then the autono-mous variational equation

u Df u n,n n1 ( ) Zξ= ∈+ (17)

has an exponential dichotomy on Z with associated projectors Qs,u. Without loss of generality, we may assume the same parameters K , s u,( ¯ ¯ )α as in (A4). Moreover, the roughness theorem 20 applied to (16) and (17) on intervals Z( ]−∞ ∩−n, and n ,[ ) Z∞ ∩+ with n n,− − + sufficiently large shows that the projectors satisfy

¯ → → ±∞P Q nas .ns u s u, ,

This fact will be essential in deriving theorem 4 below.

3.1. The principal part of the homoclinic orbit and a particular pseudo-orbit

To prove our main results, it is important to fix the ‘principal part’ of the homoclinic orbit. For each N N∈ we choose a position a N( ) Z∈ such that the interval [a(N) + 1, a(N) + N − 1] contains the principal part of the homoclinic orbit xZ from (A2). More formally, we define

Z Z( ) ∥ ¯ ∥ ⩽ ∥ ¯ ∥ [ ] [ ]

⎪ ⎪

⎧⎨⎩

⎫⎬⎭

ξ ξ= ∈ − − ∀ ∈∉ + + − ∉ + + −

b N b x x c: min : sup supn b b N

nn c c N

n1, 1 1, 1

and

Z

Z( )( ) ¯

¯ ⎜ ⎟

⎧⎨⎪

⎩⎪⎛⎝⎢⎣⎢

⎥⎦⎥⎞⎠

ξ

ξ=

≠ ∈

= ∈a N

b N x n

bN

x n:

, if for all ,

2, if for some .

n

n

The latter case occurs if the fixed point ξ has several pre-images and one of these pre-images coincides with an orbit point. If the fixed point has no stable eigenvalue, then this orbit is called a snap-back repeller in the literature, see [26]. An example of this kind is discussed in section 6.4. The extra condition in the snap-back case guarantees the convergence of the dichotomy projectors at the left and right boundaries, see (28).

Figure 2 illustrates this choice of intervals in both cases.From condition (A2) we deduce the first assertion of the following lemma.


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Lemma 2. Assume (A1)–(A4).

(i) For each 0ε> there exists an N N∈ such that xa N n∥ ¯ ∥ ⩽( ) ξ ε−+ for all n 0⩽ and for all n N⩾ .

(ii) For any N1 N∈ there exists an N N2 1> such that

[ ( ) ( ) ] [ ( ) ( ) ]+ + − ⊂ + + −a N a N N a N a N N1, 1 1, 1 .1 1 1 2 2 2 (18)

Proof. (of (ii)) First note that ε ξ= − >− ∉ − + − xsup 0n a N a N N n1, 11 1 1∥ ¯ ∥[ ( ) ( ) ] , since xn¯ is noncon-

stant. Moreover, let ε ξ= −+ + −xa N N 11 1∥ ¯ ∥( ) . If 0ε =+ , then we are in the snap-back case and

set 1

2ε ε= −, otherwise, we define min ,1

2 ε ε ε= − + . Applying assertion (i) to this ε yields a

number N N N2 1= > with property (18).

Let J n n,[ ]= − + , where the cases = −∞−n and n = ∞+ are allowed and fix an N N∈ . Given a sequence of symbols sJ A N

J( )∈ Σ , we define the corresponding pseudo-orbit s N,J J( )ξ

as follows

⎧⎨⎩

ξξ

==

∈+

s Ns

xn J, :

, if 0,, otherwise,

.n Jn

a N sn

( )

¯ ( ) (19)

In what follows, we study the variational equations obtained by linearizing at orbits from the ε-neighborhoods

s N x x s N, : : , .d( ( )) ( ) ∥ ( )∥ ⩽ RZ Z ZZ

Z Z Zξ ξ ε= ∈ −ε ∞B

3.2. Uniform dichotomies of multi-humped pseudo-orbits

In the following, we concatenate two orbits, each of which has exponential dichotomies, but for which the projectors at the interface do not coincide. This situation will be repaired by modifying the system matrix at the interface according to the following lemma.

Lemma 3. Consider B d d,R∈ and three projectors P P Q, , d d1 0 0

,R∈− such that

BP P B.1 0=− (20)

Then the perturbed matrix

B I B Q P P Q, 20 0 0 0 0( )= + ∆ ∆ = − −

satisfies

B P Q B0 1 0 0=− (21)

Figure 2. Choice of a(N ).


3356

and the estimate

Q P P Q .0 0 0 0∥ ∥ ⩽ (∥ ∥ ∥ ∥)∥ ∥∆ + − (22)

If 1∥ ∥∆ < and B I P I P: 1 0( ) → ( )− −−R R is invertible, then so is

B I P I Q:0 1 0( ) → ( )− −−R R and

B B1

1.I P I P0

1 11 1∥( ) ∥ ⩽

∥ ∥∥( ) ∥( ) ( )

− ∆| −

−| −

−− −R R (23)

Proof. A direct computation shows

I Q P I Q I P I Q P Q Q P P .0 0 0 0 0 0 0 0 0 0( )( ) ( ) ( )+ ∆ = + − − = + − + − (24)

From the first equality and (20), we obtain

B P I P B Q P B Q I B Q B ,0 1 0 0 0 0 0 0( ) ( )= + ∆ = = + ∆ =−

while the second equality leads to estimate (22). If 1∥ ∥∆ < , it is clear that I + ∆ is invertible

with I 1 1

1∥( ) ∥ ⩽

∥ ∥+ ∆ −

− ∆. Moreover, equation (24) shows that I + ∆ maps I P0( )−R into

I Q0( )−R and P0( )R into Q0( )R , so that both projectors must have the same rank and I + ∆ is a bijection between their ranges. We conclude that B I B I P I Q:0 1 0( ) ( ) → ( )= + ∆ − −−R R is invertible and satisfies estimate (23).

Our main result on the persistence of exponential dichotomies is the following theorem.

Theorem 4. Let conditions (A1)–(A4) be satisfied and let 0 s u s u, ,¯α α< < .

(i) There exists an NED N∈ such that for all N NED⩾ and for all s A N( )ZZ∈ Σ the variational

equation

u Df s N u n, ,n n n1 ( ( )) ZZξ= ∈+ (25)

has an exponential dichotomy with constants K16 , s u3

,( ¯ )α . (ii) Fix 0ε> . Then there exists an N NED¯ ⩾ such that for all N N⩾ ¯ the dichotomy projectors

Pns u, , n Z∈ of (25) satisfy the inequalities for n Z∈

∥ ∥ ⩽ ¯ ∥ ¯ ∥ ⩽ ¯ ( )

εε

− =− = ∈ … −+

P Q s

P P s k N

, if 0,

, if 1, , 1 .ns u s u

n

ns u

a N ks u

n

, ,

, ,

(iii) Let 0 s u s u, ,β α< < . Then there exists an 0EDε > such that the difference equation

u Df x d u nd ,n n n n10

1( ) Z∫ τ τ= + ⋅ ∈+ (26)


,( ¯ )β for all +x x,Z Z ( ( ))ξ∈Z Z ZBεd s N,ED and for all N NED⩾ and Z

Z( )∈ Σs A N .

Remark 5. In the case dn = 0 for all n Z∈ , theorem 4 (iii) proves a uniform exponential dichotomy of the variational equation

u Df x u n,n n n1 ( ) Z= ∈+

for all pseudo-orbits x s N,ED( ( ))Z Z Zξ∈ εB .


3357

A related result for invertible maps is proved by Palmer in [33, theorem 2.1]. An expo-nential dichotomy on Z holds if the difference equation has uniform dichotomies on all finite intervals [b, b + M] with b from a suitable, relatively dense set of Z.

Proof. The proof of (i) is based on the following idea. We begin with exponential dichoto-mies on the subintervals marked in red and black in figure 3. Then, we invoke the repair mechanism from lemma 3 with matrices ∆± in order to combine these dichotomies into a di-chotomy on Z. Finally, application of the roughness theorem 20 proves the desired dichotomy of the original equation.

Step 1: Choice of constants. Fix 0 s u s u s u, , ,ˆ ¯α α α< < < and choose N0 N∈ such that

K4 e 1.N2 s u s u, , 0¯ ( ¯ ˆ ) <α α− − (27)

Define for N N∈ the matrices

N P Q Q P N Q P P Q: 2 , : 2 ,a Ns s s

a Ns s

a N Ns

a N Ns s

1 1( ) ¯ ¯ ( ) ¯ ¯( ) ( ) ( ) ( )∆ = − − ∆ = − −− + + + + +

where Qs,u are the dichotomy projectors of (17). Then, it follows from lemma 3, equation (22) that

N K P Q N K P Q2 , 2 ,a Ns s

a N Ns s

1∥ ( )∥ ⩽ ¯ ∥ ¯ ∥ ∥ ( )∥ ⩽ ¯ ∥ ¯ ∥( ) ( )∆ − ∆ −− + + +

and these quantities converge to 0 as N → ∞. Choose N1 with

∆ ∆− +N N N Nmax ,1

2for all 1∥ ( )∥ ∥ ( )∥ ⩽ ⩾ (28)

and N2 such that

N Df N Df x Kmax , 8 , ,a N N s u s u13

, ,∥ ( ) ( )∥ ∥ ( ) ( ¯ )∥ ⩽ ( ¯ ˆ )( )ξ γ α α∆ ∆− + + − (29)

holds for all N N2⩾ . Here, γ is the function from the roughness theorem 20. Finally, we define N N N N: max , ,ED 0 1 2 = .

Step 2: Definition of the repaired system. Let N NED⩾ and s A N( )ZZ∈ Σ . We define for n Z∈

the matrices A Df s N: ,n n( ( ))Zξ= and

B

I N Df s

I N Df x s N

Df s N

:

, if 1,, if 1,

, , otherwise.n

n

a N N n

n

1

1

( ( )) ( ) ( ( )) ( ¯ )

( ( ))( )

Z

⎧⎨⎪

⎩⎪

ξ

ξ=

+ ∆ =+ ∆ = −

− +

+ + −

Using (29), we conclude

A BN Df s

N Df x s N

K

, if 1,, if 1,

0, otherwise

8 , , .

n n

n

a N N n

s u s u

1

1

3, ,

∥ ∥∥ ( ) ( )∥ ∥ ( ) ( ¯ )∥

⩽ ( ¯ ˆ )

( )

⎧⎨⎪

⎩⎪

ξ

γ α α

− =∆ =∆ = −

− +

+ + −

Figure 3. Illustration of the repair mechanism.


3358

The roughness theorem 20 yields result (i), i.e. an exponential dichotomy with constants K16 , s u

3,( ¯ )α of the original system (25), if the repaired system

u B u n,n n n1 Z= ∈+ (30)


,( ¯ ˆ )α .Step 3: System (30) has an exponential dichotomy on Z with constants K8 , s u

3,( ¯ ˆ )α . We

introduce for n Z∈ the projectors

PQ s

P s k N:

, if 0,

, if 1, , 1 .ns u

s un

a N ks u

n

,,

,ˆ

¯ ( )

⎧⎨⎩

=== ∈ … −+

(31)

One immediately sees from equation (21) in lemma 3 that these projectors satisfy the invari-ance condition (i) in definition 18 w.r.t. the difference equation (30).

By the assumptions, system (30) has exponential dichotomies on every subinterval (red or black in figure 3) with constants K , s u,( ¯ ¯ )α . Our construction shows that every subinter-val can be enlarged by its right neighbor, where the dichotomy constant grows at most by I N 2∥ ( )∥ ⩽+ ∆± due to (28). Therefore, we have exponential dichotomies on the extended

subintervals with projectors from (31) and constants K2 , s u,( ¯ ¯ )α .Next, we introduce the index set

k s: 1 ,k1 Z Z= ∈ =

and the successor and the predecessor of each k 1Z∈ by

k k p k kinf : , sup : ,1 1( ) ( ) Z Zν = ∈ > = ∈ <

where k( )ν = ∞ resp. ( ) = −∞p k if the corresponding sets are empty.Denote by Φ the solution operator of (30). We prove the dichotomy estimate in forward

time. For arbitrary n m⩾ ,let

k m n kn s

m n: min , , :

, if 0,max , , otherwise.

n1

1[ ]

[ ] Z Z⎧⎨⎩

= ∩ ==

∩− +

It follows from (27) that

∏

∏

∏

∏

ν

Φ = Φ Φ Φ

Φ Φ Φ

⋅

⋅ ⋅

α α α ν

α α α ν α ν

α α ν

α

+ + − −

+ −∈ ∩

− − − −

∈ ∩

− −

− − + −

∈ ∩

− − − − −

− − + −

∈ ∩

− −

− −

+ −

+

− +

+ −

− +

+ −

− +

+ −

− +

n m P n k P k k P k m P

n k P k m P k k P

K K K

K K

K

K

, , , ,

, , ,

2 e 4 e 4 e

8 e 4 e e

8 e e

8 e .

ms

ks

ks

ms

ks

ms

k k p kks

n k k m

k k p k

k k

n k k m

k k p k

k k k k

n k k m

k k p k

k k

n m

,

2

,

2

3

,

2

3

,

3

s s s

s s s s

s s

s

1

1

1

1

Z

Z

Z

Z

∥ ( ) ˆ ∥ ∥ ( ) ˆ ( ) ˆ ( ) ˆ ∥

⩽ ∥ ( ) ˆ ∥∥ ( ) ˆ ∥ ∥ ( ( ) ) ˆ ∥

⩽ ¯ ¯ ¯

⩽ ¯ [ ¯ ]

⩽ ¯

⩽ ¯

[ ( )]¯ ( ) ¯ ( )

[ ( )]

¯ ( ( ) )

¯ ( )

[ ( )]

( ¯ ˆ )( ( ) ) ˆ ( ( ) )

¯ ( )

[ ( )]

ˆ ( ( ) )

ˆ ( )


3359

In a similar fashion, one obtains for n m⩾

m n P K, 8 e .nu n m3 u∥ ¯ ( ) ˆ ∥ ⩽ ¯ ˆ ( )Φ α− −

Thus, (30) has an exponential dichotomy on Z with data K P8 , ,s u ns u3

,,( ¯ ˆ ˆ )α , and the proof of

(i) is complete.To prove (ii), choose an arbitrary 0ε> and let K8 , ,s u s u

3, ,( ¯ ˆ )µ µ α α= be the function from

the roughness theorem 20. Then, one finds an N NED¯ ⩾ such that

N Df N Df xK

N Nmax ,16

for all .a N N 1 3∥ ( ) ( )∥ ∥ ( ) ( ¯ )∥ ⩽ ¯

¯ ⩾ ¯( )ξεµ

∆ ∆− + + −

The roughness theorem 20 proves an exponential dichotomy for (25) with data K16 ,3( ¯ P,s u n

s u,

, )α . Estimate (A.5) reads

P P K N Df N Df x

KK

16 max ,

1616

.

ns u

ns u

a N N, , 3

1

33

∥ ˆ ∥ ⩽ ¯ ∥ ( ) ( )∥ ∥ ( ) ( ¯ )∥

⩽ ¯ ¯¯ ¯

( )µ ξ

µεµ

ε

− ∆ ∆

=

− + + −

Using definition (31), our estimates follow.To prove (iii), we recall the compact hyperbolic set H from (2) and choose Ω compact and

sufficiently large such that xdist , 1( ) ⩽H implies x ∈ Ω. Further, we introduce the modulus of continuity

Df Df x Df y x y x y, , sup : , , .( ) ∥ ( ) ( )∥ ∥ ∥ ⩽ ω δ δΩ = − ∈ Ω − (32)

Fix N NED⩾ and 0 s u s u, ,β α< < . Choose EDε such that

Df x d Df s N Df Kd , , , 16 , ,n n n s u s u0

1

ED3

, ,( ) ( ( )) ⩽ ( ) ⩽ ( ¯ )Z∫ τ τ ξ ω ε γ α β+ − Ω

for all s A N( )ZZ∈ Σ and for all x x d s N, ,ED( ( ))Z Z Z Z Zξ+ ∈ εB . Then, the roughness theorem 20

applies and proves an exponential dichotomy of (26) with constants K32 , s u3

,( ¯ )β .

In the next section we need a result that is analogous to theorem 4 for finite but sufficiently large intervals J Z⊂ . The proof uses the same techniques and shows that the resulting data NED, EDε and N N¯ ¯ ( ¯)ε= as well as the dichotomy constants can be taken independently of J. For completeness, we state the theorem.

Theorem 6. Let conditions (A1)–(A4) be satisfied and let 0 s u s u, ,¯α α< < . Then, there exists an 00 > such that assertions (i)–(iii) of theorem 4 hold for all intervals J n n,[ ]= − + with n n 0⩾−+ − .

4. Multi-humped orbits on finite and infinite intervals

In this section, we prove the existence of a multi-humped homoclinic orbit w.r.t. each finite or infinite coding sequence. In all the considerations of the following sections, we fix the para-meters 0 s u s u, ,¯α α< < and 0 s u s u, ,β α< < .


3360

4.1. Solutions of boundary value problems

Let J n n,[ ]= − + , J n n, 1˜ [ ]= −− + be a finite interval or let J J Z= = . We define the operator

xx f x n Jb x x J

, ,, , if is finite

.J Jn n

n n

1( ) ( ) ˜( )

⎛

⎝⎜

⎞

⎠⎟Γ =

− ∈+

− +

We impose the following condition on the boundary operator.

(A5) The operator b satisfies b ,d d1 2( )R R∈ C and b , 0( )ξ ξ = . Furthermore, the linear map

B D b D b Q Q: , , :Q Qs u d

1 2s u( )( ) ( ) ( ) ( ) →( ) ( ) Rξ ξ ξ ξ= ⊕| | R RR R

is invertible, where Qs,u are the dichotomy projectors from (17).

Examples of boundary operators that satisfy (A5) are

• periodic boundary conditions b x y x y, :per( ) = − and

• projection boundary conditions b x yY x

Y y, : s

T

uTproj( )( )( )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

ξ

ξ=

−

−, where Ys,u are bases of

Qu s,( ( ))⊥R , respectively.

Now, we have all the tools at hand to prove our main existence theorem for finite and infi-nite homoclinic orbits.

Theorem 7. Assume (A1)–(A5). Then there exists a constant solε and for any 0 sol⩽ε ε< a number Nε, such that for all N N⩾ ε the following property holds true for both J Z= and J finite with J N 1⩾| | + .

For any sJ A NJ

( )∈ Σ , the operator JΓ has a unique zero x s N,J J J( ( ))ξ∈ εB , where the pseudo- orbit s N,J J( )ξ is defined in (19). Furthermore, s N s N, ,J J J J

1 2( ( )) ( ( ))ξ ξ∩ = ∅ε εB B for any s sJ J A N

J1 2( )≠ ∈ Σ .

Proof. Step 1: Choice of constants. Theorems 4 and 6 give the constants EDε and NED, such that the dichotomy rates of the variational equation w.r.t. the pseudo-orbit s N,J J( )ξ , N NED⩾ do not depend on the specific sequence sJ. As a consequence, we obtain uniform estimates for the corresponding Green function, defined in (A.3) with K K16 3¯= , n J∈ :

G n m K K

K C

, e e

1

1 e

e

1 e: .

m J m

nn m

m n

m n

G

1

s u

s

u

u

( ) ⩽ ( ) ( )

⎜ ⎟⎛⎝

⎞⎠

∑ ∑ ∑+

=−

+−

=

α α

α

α

α

∈ =−∞

− −

= +

∞− −

−

−

−

In the following, let C C 1B b1 = =− if J Z= , and define C B: max , 1B1

1 ∥ ∥ = −− with B

from (A5), C D b D b: max , , , , 1b 1 2∥ ( )∥ ∥ ( )∥ ξ ξ ξ ξ= if J is finite.

Applying assertion (ii) of theorem 6 with 1

2ε = , we find an N N0 ED⩾ , such that system (25)

has an exponential dichotomy on the interval J n n,[ ]= − + for all N N0⩾ and sJ A NJ

( )∈ Σ . The

corresponding projectors Pns u, depend on sJ, but satisfy uniform estimates. Note that sJ A N

J( )∈ Σ

and hence, s s 0n n= =− + and Q Psns 1

2∥ ∥ ⩽−

− and Q Pu

nu 1

2∥ ∥ ⩽−

+ because of theorem 6 (ii).


3361

We define

T I Q P P Q

T I Q P P Q

: : ,

: :ns s

ns

ns s

nu u

nu

nu u

( ) → ( )( ) → ( )

= + −= + −

− − −

+ + +

R R

R R

and observe that these matrices are invertible with uniformly bounded inverse T 2ns 1∥( ) ∥ ⩽−

−,

T 2nu 1∥( ) ∥ ⩽−

+. Let ( )= − +B A AJ n n with

A D b T D b n n T

A D b T D b n n T

, , , ,

, , , .

n ns

Q ns

Q

n nu

Q nu

Q

11

21

21

11

s s

u u

( )( ) ( ) ( )( )

( )( ) ( ) ¯ ( )( )( ) ( )

( ) ( )

ξ ξ ξ ξ

ξ ξ ξ ξ

= + Φ

= + Φ

|−

+ − |−

|−

− + |−

− − −

+ + +

R R

R R

The second terms decay exponentially as → ±∞±n because of the uniform dichotomy rates, while the first terms converge to D b , Q1 s( ) ( )ξ ξ |R and D b , Q2 u( ) ( )ξ ξ |R , respectively, as

→ ±∞±n . Thus, there exists N N1 0⩾ , such that for all N N1⩾ the estimate B BJ C

1

2B 1

∥ ∥ ⩽−−

holds.Let KC C C17 B b G

11( )η = −

− . Using the modulus of continuity from (32), there exists an ε such that

Df

Db y z Db y z

, ,2

,

, ,2

for all , .

( ˜) ⩽

∥ ( ) ( )∥ ⩽ ( )˜

ω εη

ξ ξη

ξ

Ω

− ∈ εB (33)

Choose 0ε> such that

x xmax min 2 .n a N a N N m m n

n m1, 1 ,ED ED ED

∥ ¯ ¯ ∥ ˆ[ ( ) ( ) ] Z

ε− >∈ + + − ∈ ≠ (34)

Fix min , ,sol ED ˜ ˆε ε ε ε= and let sol⩽ε ε .Using lemma 2 (i), we find an N N1⩾ε such that we obtain for all N N⩾ ε

∥¯ ∥ ⩽ ⩽ ⩾( ) ξηε−+x n n N

2for all 1 and .a N n (35)

Fix N N⩾ ε, let J be an interval, J N 1⩾| | + , and choose an arbitrary sequence of symbols

sJ A NJ

( )∈ Σ . By theorems 4 and 6, the variational equation along the corresponding pseudo-orbit s N,J J( )ξ has an exponential dichotomy with data K P, ,s u n

s u,

,( )α .Step 2: D s N,J J J( ( ))ξΓ has a uniformly bounded inverse. For yJ J˜ ˜∈ S and r dR∈ , solve the

boundary value problem

( )( ( )) ( ( )) ˜

( ) ( )˜

⎧⎨⎩

ξξ

ξ ξ ξ ξΓ = ⇔

− = ∈+ =

+

− +

D s N uyr

u Df s N u y n J

D b u D b u r,

, , ,

, , ,J J J J

J n n J n n

n n

1

1 2

(36)

where the second row vanishes if J Z= . Here, JS denotes the Banach space of the bounded sequences on J, see appendix A. The general solution of the linear equation is given by

u n n v n n v G n m y v P v P, , , 1 , , .nm J

m ns

nu( ) ¯ ( ) ( ) ( ) ( )

˜∑= Φ + Φ + + ∈ ∈− − + +

∈− +− +R R

(37)

In the case J Z= , the first two terms are equal to 0 and the third term defines the unique bounded solution on Z. In the finite case, we plug (37) into the boundary condition in (36) and in this way guarantee the uniqueness of the solution. Then, we transform the system into


3362

n±-independent spaces by introducing the variables v T vns¯ =− −−

and v T vnu¯ =+ ++

. In the new coordinates, the resulting system reads

Bvv R,J( )¯¯ =−+

where

∑ ∑ξ ξ ξ ξ= − + − +

+∈

−∈

+

∞

R r D b G n m y D b G n m y

R r C C y

, , 1 , , 1 ,

2 .m J

mm J

m

b G J

1 2( ) ( ) ( ) ( )

∥ ∥ ⩽ ∥ ∥ ∥ ∥˜ ˜

˜

Since B BJ C

1

2B 1

∥ ∥ ⩽−−

, the Banach lemma applies and guarantees that BJ is invertible and

B C2J B1

1∥ ∥ ⩽−− . Furthermore, the following estimates hold true:

v v C r C C C y2 4 8 .B B b G J1 1∥ ∥ ⩽ ∥ ¯ ∥ ⩽ ∥ ∥ ∥ ∥˜+± ± ∞− −

This implies that for all n J∈ ,

u KC r KC C C y C y

KC C C r y y r

8 16

17

n B B b G J G J

B b G J J1

1 1

1

∥ ∥ ⩽ ∥ ∥ ∥ ∥ ∥ ∥⩽ (∥ ∥ ∥ ∥ ) (∥ ∥ ∥ ∥)

˜ ˜

˜ ˜η

+ +

+ = +∞ ∞

∞−

∞

− −

−

and consequently

D s N, .J J J1 1∥ ( ( )) ∥ ⩽ξ ηΓ − −

Step 3: The existence of a unique solution in s N,J J( ( ))ξεB . We apply the Lipschitz inverse mapping theorem 21 with the setting

Y Z F y s N, , , max , , , ,J Jd

J J J0( ∥ ∥ ) ( ∥ ∥ ∥ ∥) ( )˜ R ξ= ⋅ = × ⋅ ⋅ = Γ =∞ ∞S S

and in the case J Z= we define Z ,( ∥ ∥ )Z= ⋅ ∞S . Finally, we choose δ ε= and 2

κ = η.Assumption (B.1) is satisfied, since by (33) we find that

D z D s N Df z Df s N

Db z z Db

, max sup , ,

, ,2

J J J J Jn J

n n J

n n

∥ ( ) ( ( ))∥ ⩽ ∥ ( ) ( ( ))∥

∥ ( ) ( )∥ ⩽

˜ξ ξ

ξ ξη

Γ − Γ −

−

∈

− +

holds for all z s N,J J J( ( ))ξ∈ εB .Assumption (B.2) follows from (35):

s N s N f s N

x x

, , ,

max ,2

.

J J J n J n J n J

a N a N N

1

1

∥ ( ( ))∥ ∥( ( ) ( ( ))) ∥

∥ ¯ ∥ ∥ ¯ ∥ ⩽

˜

( ) ( )

ξ ξ ξ

ξ ξηε

Γ = −

= − −

+ ∈ ∞

+ +

Thus, theorem 21 proves that JΓ has a unique zero x s N,J J J( ( ))ξ∈ εB .

Step 4: s N s N, ,J J J J1 2( ( )) ( ( ))ξ ξ∩ = ∅ε εB B for any s sJ J A N

J1 2( )≠ ∈ Σ . From (34), we conclude

that there is an N1, 1ED[ ]∈ − such that


3363

ξ ξ∩ = ∅ = ≠ε εs N s N n s s, , for all satisfying , .n J n J n n1 2 1 2 B B( ( )) ( ( )) (38)

Let s sJ J A NJ1 2

( )≠ ∈ Σ . Then, there exists an m J∈ such that s s k N0 1, 1m m1 2 [ ]= ≠ = ∈ − . It

follows that s s k k km kk

mk2 2( ) ( )σ σ= = = + − =+ −

− − . On the other hand, sm k

1 ≠+ − ,

since sm k1 =+ − implies that s s km

km k

k1 1( ) ( )σ σ= = =−+ −

−

, a contradiction. Using (38), the claim follows.

4.2. Exponential decay of errors

Precise error estimates are based on the fact that all the orbits in s N,J J( ( ))ξεB converge expo-nentially fast towards each other in the interior of J at a uniform rate.

Theorem 8. Assume (A1)–(A4), then the following statement holds for all 0 ED⩽ε ε< , N NED⩾ , J N 1⩾| | + .

Any two finite orbit segments z z s N, ,J J J J1 2 ( ( ))ξ∈ εB satisfy the estimate

z z L n J L Ke e , , : 64n nn n n n1 2 3s u∥ ∥ ⩽ ( ) ¯( ) ( )ε− + ∈ =β β− − − −− + (39)

with the constants N,ED EDε and s u,β from theorem 6 (iii).

Proof. For z z s N, ,J J J J1 2 ( ( ))ξ∈ εB , define d z zJ J J

1 2= − . Then, dJ solves the difference equation

d z z f z f z Df z d d

A d n J

d

, .

n n n n n n n n

n n

1 11

12 1 2

0

12( ) ( ) ( )

˜∫ τ τ= − = − = + ⋅

= ∈

+ + +

Note that z d s N,n n n J2 ( ( ))τ ξ+ ∈ εB for all n J∈ and all 0, 1[ ]τ∈ . Thus, theorem 6 (iii) applies

and yields an exponential dichotomy of the difference equation

u A u n J,n n n1 ˜= ∈+ (40)

with uniform constants K32L

23¯= , s u,β and projectors Qn

s u, . Denote by Φ the solution operator

of (40). Since dJ is a solution of (40), we conclude

d n n Q d n n Q d n J, , ,n ns

n nu

n( ) ¯ ( )= Φ + Φ ∈− +− − + +

and

dL

dL

d

L2

e2

e

e e .

nn n

nn n

n

n n n n

s u

s u

∥ ∥ ⩽ ∥ ∥ ∥ ∥

⩽ ( )

( ) ( )

( ) ( )ε

+

+

β β

β β

− − − −

− − − −

−−

++

− +

For a given sequence sJ A NJ

( )∈ Σ , the choice of a boundary condition in (A5) leads to a unique zero zJ

1 of JΓ in s N,J J( ( ))ξεB , see theorem 7. Changing the boundary condition results in an alternative orbit segment z s N,J J J

2 ( ( ))ξ∈ εB . The error estimate (39) shows that the bound-ary condition is only influential over a short range. For a similar observation in nonautono-mous systems, we refer to [19, theorem 5].

Combining theorems 7 and 8, we immediately obtain the following approximation result for multi-humped homoclinic orbits.


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Corollary 9. Assume (A1)–(A5). Let 0 sol⩽ε ε< and Nε be given as in theorem 7. Consider a

finite interval J n n,[ ]= − + with J N 1⩾| | + and N N⩾ ε. For any sequence of symbols s A N( )ZZ∈ Σ

such that s s 0n n= =− + , denote by s N,( )Z Zξ the pseudo-orbit defined in (19).Then there exist unique orbits x s N,( ( ))Z Z Zξ∈ εB and y s N,J J J( ( ))ξ∈ εB , satisfying

x y0, 0,J J( ) ( )Z ZΓ = Γ =

respectively. Furthermore, approximation errors can be estimated as

x y L n Je e , .n nn n n ns u∥ ∥ ⩽ ( )( ) ( )ε− + ∈β β− − − −− +

4.3. Hausdorff distance between finite and infinite orbits

Choose 0 sol⩽ε ε< , N N⩾ ε as in theorem 7. We define the set of all the infinite and finite multi-humped orbits for sufficiently large finite intervals J, J N 1⩾| | + :

N x s x s N x

N x s x s N x

, : : , , 0 ,

, : : , , 0 .

J JJ

J A NJ

J J J J J

A N

( ) ( ( )) ( )

( ) ( ( )) ( ) ( )

( )ZZ

ZZ

Z Z Z Z Z

ε ξ

ε ξ

= ∈ Ω ∃ ∈ Σ ∈ Γ =

= ∈ Ω ∃ ∈ Σ ∈ Γ =

ε

ε

X B

X B

(41)

Theorem 10. Assume (A1)–(A5). Fix 0 sol⩽ε ε< , N N⩾ ε as in theorem 7. For any 0δ> there exists an n 0¯ > such that for all J n n,[ ]= − + with n n n, ⩾ ¯− − + and n n N3 1⩾− ++ − the fol-lowing assertion holds

N Ndist , , .J JH( ( ) ( )) ⩽ε ε δ−|X X

Here, N , J( )ε |X is the set of sequences from N ,( )εX , restricted to the finite interval J and distH is the Hausdorff distance (11) with the dist-function defined in (12).

Proof. Let Ω be the compact neighborhood of the homoclinic orbit from theorem 4 and let

C diam( )= Ω . Further, denote by K32 ,Ls u2

3,( ¯ )β= the uniform dichotomy data, which theo-

rems 4, 6 (iii) provide for the variational equations along any orbit x N ,( )Z ε∈X . For fixed 0δ> , we choose n such that

( )

( ) ⩽

\ ˆ ˆ( ) ( )

( ) ( )

⎧⎨⎩

⎫⎬⎭

∑ ∑

∑

ε

ε δ

+ +

+

β β

β β

∈

−| |

∈

−| | − − − − − −

∈

−| | − − − −

− +

− +

C L

L

max 2 2 e e ,

2 e e

n J J

n

n J

n n n N n N n

n J

n n n n n

s u

s u

for J n n,[ ]= − + , J n N n N,ˆ [ ]= + −− + , n n n, ⩾ ¯− − + , n n N3 1⩾− ++ − . Note the different di-rections of exponential decay in the second and third sums.

Step 1: For any x N ,J J( )ε∈ |X there exists a y N ,J J( )ε∈X such that x ydist ,J J( ) ⩽ δ, where the

distance between two segments is defined in (12). For x N ,( )Z ε∈X , let s A N( )ZZ∈ Σ be the corresp-

onding sequence of symbols. Since its restriction to the finite interval J does not necessarily lie in N ,J( )εX , we introduce an adapted cutoff process. Let i n n n N smax , : 0n [ ] = ∈ + =− − − , i n n N n smin , : 0n [ ] = ∈ − =+ + + and define for n J∈

ss n i i, for , ,0, otherwise.n

nˆ [ ]⎧⎨⎩

=∈ − +


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For the sequence sJ, the existence of an orbit y N ,J J( )ε∈X follows from theorem 7. The esti-mates for the difference of these orbits come from theorem 8:

x y L n J

x y C n J J

e e , ,

, .

n nn n N n N n

n n

s u∥ ∥ ⩽ ( ) ˆ

∥ ∥ ⩽ \ ˆ

( ) ( )ε− + ∈

− ∈

β β− − − − − −− +

Combining these estimates, we obtain

x y x y

C L

dist , 2

2 2 e e .

J Jn J

nn n

n J J

n

n J

n n n N n N ns u

( ) ∥ ∥

⩽ ( ) ⩽\ ˆ ˆ

( ) ( )

∑

∑ ∑ε δ

= −

+ +β β∈

−| |

∈

−| |

∈

−| | − − − − − −− +

Step 2: For any x N ,J J( )ε∈X there exists a y N ,J J( )ε∈ |X such that x ydist ,J J( ) ⩽ δ. Let

x N ,J J( )ε∈X and let sJ A NJ

( )∈ Σ be the corresponding symbolic sequence. Define for n Z∈

ss n J, for ,0, otherwise.n

n˜ ⎧⎨⎩=

∈

Since s s 0n n= =− + , this setting leads to s A N˜ ( )Z∈ Σ . Through theorem 7, we obtain an orbit

y N ,( )Z ε∈X , which belongs to the sequence sZ, and theorem 8 gives the estimate

x y L n Je e , .n nn n n ns u∥ ∥ ⩽ ( )( ) ( )ε− + ∈β β− − − −− +

Thus, we obtain

x y Ldist , 2 e e .J Jn J

n n n n ns u( ) ⩽ ( ) ⩽( ) ( )∑ε δ+β β

∈

−| | − − − −− +

5. The homoclinic theorem for finite and infinite orbits in noninvertible systems

In this section, we show how the homoclinic theorem for noninvertible maps [42, 22] can be derived in a straightforward manner from the results of section 4. In addition, we prove an analogous theorem for finite orbit segments of sufficient length. The idea is to first prove the conjugacy of a subshift with the map F (see (4)), induced by f on the solution set N ,( )εX from (41). Then one shows that the solution set coincides with the orbit set Orb( )O from (3) for a properly chosen neighborhood O. This approach will then be transferred to finite orbit seg-ments J for a suitably defined operator JF , see (10).

5.1. Conjugacy between shift-dynamics and the dynamics on X( )εN,

Instead of directly considering the maximal invariant set Orb( )O , it is advantageous to start with the set N ,( )εX , defined in (41) with sol⩽ε ε and N N⩾ ε. Exploiting the results from the previous section, we prove the existence of a homeomorphism h, which conjugates the sub-

shift σ on A N( )ZΣ to the orbit shift F on N ,( )εX , see (4):


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(42)

Theorem 11. Assume (A1)–(A4). Let 0 sol⩽ε ε< , N N⩾ ε, as in theorem 7. Then there exists a homeomorphism h, such that the diagram (42) commutes, i.e. h h σ= F .

Proof. Fix 0 sol⩽ε ε< and N N⩾ ε.

Step 1.: Definition of h N: ,A N → ( )( )Z εΣ X . It follows from theorem 7 that for any s A N( )Z

Z∈ Σ , there exists a unique x s s N,( ) ( ( ))Z Z Z Zξ∈ εB such that x s 0( ( ))Z Z ZΓ = . We define

h s x s s: , .A N( ) ( ) ( )Z Z Z ZZ= ∈ Σ

Step 2: h is invertible. By definition, for each x N ,( )Z ε∈X there exists an s A N( )ZZ∈ Σ

such that h s x( )Z Z= . Furthermore, theorem 7 guarantees the uniqueness of sZ, since

B s N B s N, ,( ( )) ( ( ˜ ))Z Z Z Zξ ξ∩ = ∅ε ε for any s s A N˜ ( )Z ZZ≠ ∈ Σ .

Step 3: h and h−1 are continuous. Let C diam( )= Ω and K32L

23¯= , s u,β be the uniform

dichotomy constant for the variational equation along each x N ,( )Z ε∈X , see theorem 4 (iii).For any fixed 01δ > , careful estimates show there exist integers n n0< <− + such that the

following estimates hold for J n n,[ ]= − + ,

C L2 2 e e .n J

n

n J

n n n n n1

s u( ) ⩽\

( ) ( )

Z∑ ∑ε δ+ +β β

∈

−| |

∈

−| | − − − −− +

Choose 2 n n2

max , δ < − − − + . For all s A N1,2

( )ZZ∈ Σ satisfying

d s s s s, 2n

nn n

1 2 1 22( ) ⩽Z Z

Z∑ δ= | − |∈

−| |

we conclude that s sn n1 2= for all n J∈ . Applying theorem 8 on the interval J with the boundary

operator bproj gives the estimate

h s h s L n Je e for .n nn n n n1 2 s u∥ ( ) ( ) ∥ ⩽ ( ) ( ) ( )

Z Z ε− + ∈β β− − − −− +

Inserting this into the dist-function and applying a global estimate outside the finite interval J, we obtain

h s h s C Ldist , 2 2 e e .n J

n

n J

n n n n n1 21

s u( ( ) ( )) ⩽ ( ) ⩽\

( ) ( )Z Z

Z∑ ∑ε δ+ +β β

∈

−| |

∈

−| | − − − −− +

This proves the continuity of h, and then the continuity of h−1 follows from the compactness

of A N( )ZΣ and the Hausdorff property of N ,( )εX .

Step 4: Proof of conjugacy. Let s A N( )ZZ∈ Σ . Then, x h s s N,( ) ( ( ))Z Z Z Zξ= ∈ εB . Define

y x( )Z Z= F , i.e. y f x xn n n 1( )= = + for n Z∈ and as a consequence, y s N,( ( ( ) ))Z Z Zξ σ∈ εB . The uniqueness property from theorem 7 yields the asserted conjugacy Z Z( ( ))σ = =h s y

=x h sF FZ Z( ) ( ( )).


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5.2. Conjugacy between shift-dynamics and the dynamics on O( )Orb

For sufficiently small ε and sufficiently large N, we construct an open set O, such that Orb( )O coincides with N ,( )εX . Using this result and theorem 11, the proof of the conjugacy described in diagram (5) is complete.

Theorem 12. Assume (A1)–(A4). Then there exist constants 0ε> , N > 0 and an open set O, which is a neighborhood of the homoclinic orbit, such that NOrb ,( ) ( )ε=O X .

Proof. Step 1: Choice of constants. Let solε ε= be defined as in theorem 7. Choose N N1 sol¯ ⩾+ ε such that

x i i Nfor all 0, 1,a N i2 1( ¯ ) ( ) ⩽ ⩾ ¯¯/ ( ¯ ) ¯⊂ ξ +ε ε+ +B B

see lemma 2 (i). We define

p NN x

k x x:

, if ,

, if , ,a N N

a N k a N k

1

1 1 1( ¯ )

¯ ¯ ¯ ¯

( ¯ ) ¯

( ¯ ) ( ¯ )⎪

⎧⎨⎩

ξξ ξ

=≠≠ =

+ +

+ + + + +

where the latter case accounts for possible snap-back behavior. Next, fix 2

⩽ ¯ε ε such that

x xn m n m p N

n p N m p N2 for all

, , 1, ,

1, , 1,a N n a N m1 1∥ ¯ ¯ ∥

[ ( ¯ )][ ( ¯ )] [ ( ¯ )]( ¯ ) ( ¯ )

⎧⎨⎩

ε− >≠ ∈∈ ∉+ + + +

and

f x x j p Nfor all 2, , .i N i

a N i a N j

0, 1

1 1⋃ ( ¯ ) ( ) ( ¯ ) ( ¯ )⩽ ¯ ⩽

( ¯ ) ( ¯ )

⎛

⎝⎜⎜

⎞

⎠⎟⎟ξ∪ ∩ = ∅ = …ε ε ε

++ + + +B B B

(43)

For this choice of 0ε> , we find an N N Nmax 2,⩾ ¯ + ε with Nε from theorem 7, such that a N a N N a N a N N1 1, 1 1, 1[ ( ¯ ) ( ¯ ) ¯ ] [ ( ) ( ) ]⊂+ + + + + + − , see lemma 2 (ii).

Finally, we set a N a N1( ¯ ) ( )= + − , M N p N: ( ¯ )= − and note M 1⩽ − .Step 2: Construction of O. This construction is similar to the approach in [42, theorem 5.1].We define

⋃ ξ= ∪ = = …ε ε ε+

+ + + +B x B x i p N: , : for 1, , ,i N i

a N i i a N i0

0, 1

1 1B B B( ¯ ) ( ) ( ¯ ) ( ¯ )⩽ ¯ ⩽

( ¯ ) ( ¯ )

(44)

and note that B0 ( )¯⊂ ξεB .We have to ensure that orbits in the union of these sets only enter B0 via Bp N( ¯ ) and then stay

in B0 for at least M steps. Orbits are only allowed to leave B0 via B1 and the ith iterate of an orbit point in B1 lies in Bi+1 for i p N1, , 1( ¯ )= … − . This behavior can be achieved by further shrinking the neighborhoods.

Let Vip N

i0⋃ ( ¯ )= =O , where

V B V B f V V B f V i p N: , : , : , 1, , 1.p N p N

n

Mn

i i i0 0

1

01

1⋂ ( ) ( ) ( ¯ )( ¯ ) ( ¯ )= = ∩ = ∩ = − …=

− −+

(45)

Setting (45) and condition (43) guarantee the properties

x V f x V f x V

x V f x V i p Nx V f x V n M

, ,

for 1, , 1,for 1, , .

i i

p Nn

0 0 1

1

0

( ) ( )( ) ( ¯ )( ) ( ¯ )

∈ ∉ ⇒ ∈∈ ⇒ ∈ = … −

∈ ⇒ ∈ = …+


3368

Step 3: N , Orb⊃( ) ( )εX O . For x Orb( )Z∈ O , we introduce two sequences of symbols in

A N 1( ¯ )ZΣ + and A N( )

ZΣ , respectively. Starting with A N 1( ¯ )ZΣ + , we define for n Z∈

si x V

p N j j N p N x V:

, if ,

, if 1, such that .nn i

n j p N¯

( ¯ ) [ ¯ ( ¯ )] ( ¯ )

⎧⎨⎩

=∈

+ ∈ − ∈−

This sequence is extended to N N 1¯> + symbols as follows, see table 1 for an illustration:

s

i i N s ii i s j i s

N i i M s j i s N:

, if 1, : ,1 , if 1, 1 : 0 0, 1 1,

, if 1, 1 : 0 0, 1 ,

0, otherwise.

n

n

n j n i

n j n i

[ ¯ ] ¯ [ ] ¯ [ ] ¯

¯ [ ] ¯ [ ] ¯ ¯

⎧

⎨⎪⎪

⎩⎪⎪

=

+ ∃ ∈ =+ − ∃ ∈ + = ∀ ∈ − ∧ =

+ + ∃ ∈ − − = ∀ ∈ − ∧ =+ +

− −

By theorem 7 there exists a unique y s N, 1¯ ( ( ¯ ¯ ))¯Z Z Zξ∈ +εB , satisfying y 0( ¯ )Z ZΓ = ,

and there is a second unique orbit y s N,( ( ))Z Z Zξ∈ εB , satisfying y 0( )Z ZΓ = . Since

s N s N, , 1( ( )) ( ( ¯ ¯ ))¯Z Z Z Z⊂ξ ξ +ε εB B holds true, the uniqueness of the above orbits yields y yZ Z= . Observe that x s N, 1( ( ¯ ¯ ))¯Z Z Zξ∈ +εB with x 0( )Z ZΓ = . Consequently, by the unique-ness above, x y y N ,¯ ( )Z Z Z ε= = ∈X .

Step 4: N , Orb( ) ( )⊂εX O . For any x N ,( )Z ε∈X there exists a unique s A N( )ZZ∈ Σ such that

x s N,( ( ))Z Z Zξ∈ εB and x 0( )Z ZΓ = holds true. Using the construction of O, we prove that xn ∈O for all n Z∈ , and thus x Orb( )Z∈ O , as follows.

For x Bn ip N

i1⋃ ( ¯ )∉ = we obtain x Bn 0 ⊂∈ O.For x Bn p N( ¯ )∈ the construction (44) and x N ,( )Z ε∈X show f x Bi

n 0( ) ∈ for all i M1, ,= … .

Thus, ⋂ ⊂∈ ∩ ==

−x B f V Vn p N i

M ip N1 0 O( )( ¯ ) ( ¯ ) .

For x Bn i∈ , i p N 1, , 1( ¯ )= − … , we obtain f x Vn i 1( ) ∈ + by induction, from which we deduce that x B f V Vn i i i

11( ) ⊂∈ ∩ =−

+ O.

5.3. A homoclinic theorem for finite orbits

In the section, we provide the theorem underlying the conjugacy in diagram (8). In the first step, replacing Z by J in the proof of theorem 12 and handling the boundary condition care-fully, we obtain the following finite time analog of theorem 12.

Theorem 13. Assume (A1)–(A5). Then there exist constants 0ε >X , N 0>X and an open set O, which is a neighborhood of the homoclinic orbit, such that for all intervals J with J N 1⩾| | +X it holds that J NOrb , ,J( ) ( )ε=O X X X .

Recall from (10) and (9) the finite time versions ,J JσF of the operator F and the time shift σ. With these notions the finite time analog of the homoclinic theorem, indicated by diagram (8), reads as follows.

Theorem 14. Assume (A1)–(A5) and let εX and NX be as in theorem 13. Then for any inter-

val J N 1⩾| | +X there exists a homeomorphism O→ ( )( )Σh J: Orb ,J A NJ such that diagram (8)

commutes, i.e. h hJ J J J1 σ= − F .

Proof. Step 1: Definition of hJ. Let J be a finite interval, satisfying J N 1⩾| | +X . Because

of theorem 7, for any sJ A NJ

( )∈ ΣX

there exists a unique x s s N,J J J J( ) ( ( ))ξ∈ εB XX such that x s 0J J J( ( ))Γ = . Thus, we conclude that x s N ,J J J( ) ( )ε∈X X X and consequently x s JOrb ,J J( ) ( )∈ O

by theorem 13. We define h s x s:J J J J( ) ( )= .


3369

Step 2: hJ is a homeomorphism. For any x J NOrb , ,J J( ) ( )ε∈ =O X X X , there exists—due to

theorem 7—a unique sJ A NJ

( )∈ ΣX

such that x s N,J J J( ( ))ξ∈ εB XX . Since h s xJ J J( ) = , we conclude that hJ is invertible. Note that A N

J( )ΣX

and JOrb ,( )O are finite sets with the same cardinality. Hence, hJ is a homeomorphism.

Step 3: Proof of conjugacy. For any symbolic sequence sJ A NJ

( )∈ ΣX

, we obtain x h s B s N,J J J J J( ) ( ( ))ξ= ∈ ε XX . Let y xJ J J1 ( )=− F , then it follows that y xn n 1= + for all n J 1∈ − . As a consequence, y s N,J J J J1 1( ( ( ) ))ξ σ∈ ε− −B XX and y 0J J1 1( )Γ =− − . From this, we deduce that h s y h sJ J J J J J J1 1( ( )) ( ( ))σ = =− − F .

6. Computation and visualization of the maximal invariant set and its coding

In this section, we illustrate the theory by applying the algorithm from section 2.2 to several examples. In each case we approximate the maximal invariant set cO defined in (13) by solv-ing the appropriate boundary value problems, and we indicate the symbolic coding for points close to the midpoint of the primary homoclinic orbit.

First, we comment on how to verify our hyperbolicity assumptions.

6.1. Transversality of homoclinic orbits from a geometric point of view

Unlike in the invertible case, the stable and unstable sets of a fixed point ξ are in general not immersed submanifolds of dR . Nevertheless, these sets have graph representations near the fixed point, see [12, theorem 3.1], [42, theorem 4.1]:

W x h x x U W x h x x U: , : ,ss s s s s

uu u u u uloc loc( ) ( ) ( ) ( ) ξ ξ ξ ξ= + + ∈ = + + ∈

where h U U:s u s u u s, , ,→ is smooth and Us,u denote sufficiently small neighborhoods of 0 within the stable and unstable subspaces of Df ( )ξ , respectively. Note that the global stable set can be obtained by continuation in backward time in a set valued sense, while the stable set is obtained by iterating in forward time. One can verify the transversality of the homoclinic orbit xZ, i.e. an exponential dichotomy of the variational equation (1) with projectors Pn

s u,¯ , n Z∈ , if the following conditions are satisfied (see [42, theorem 4.2]):

ξ ξ⊕ = ξ− −

|D f x T W T W D f x, is invertiblen mm x

ux

s d n mm T Wloc locm n xm

uloc

R( ¯ ) ( ) ( ) ( ¯ ) ¯ ¯ ( )¯

with

T W P T W P,xu

mu

xs

ns

loc locm n( ) ( ¯ ) ( ) ( ¯ )¯ ¯ξ ξ= =R R (46)

for all ⩽ −m Nloc and all n Nloc⩾ and a suitable Nloc N∈ . We refer to [32, proposition 5.4] and [18, theorem 3.5] for corresponding results for invertible systems in the autonomous as well as the nonautonomous case.

Assume that the variational equation (1) has an exponential dichotomy and let Φ denote its solution operator, then

P u k n u: sup , ,ns d

k n( ¯ ) ∥ ( ) ∥

⩾R= ∈ Φ < ∞R (47)

Table 1. Connection between symbolic sequences.

ss

NN N N

0 0 0 0 1 0 00 1 1 1 1 1

ZZ

… … …… − + … + + + … −


3370

see [3, theorem 2.5]. For a general point x Ws u, ( )ξ∈ , we define the stable and unstable tangent sets:

CR( ) (( ) ( )) ( ) ( ) ξ ζ ε ε ξ ζ ζ= ∈ ∃ ∈ − = =′T W v W x v: , , : 0 , 0 .xs u d s u, 1 ,

Lemma 15. Assume (A1)–(A4). Then the following assertions hold for all n Z∈ :

(i) P T Wnu

xu

n( ¯ ) ( )¯⊂ ξR , (ii) Df x Pn n

s( ( ¯ )) ( ¯ )⊂N R .

Proof.

(i) Fix n Z∈ . For ⩽ −n Nloc, the assertion follows from (46).

For > −n Nloc, observe that f x xn NN n

locloc( ¯ ) ¯=+

− and since f is noninvertible in general, we conclude that f x xn N

n Nloc

loc⊃( ¯ ) ¯ − −− in a set valued sense. As a consequence, we obtain

P Df x P Df x T W

T W .nu n N

N Nu n N

N xu

xu

locN

n

locloc loc

locloc loc

( ¯ ) ( ¯ ) ( ¯ ) ( ¯ ) ( )( )

¯

¯

ξ

ξ

= =

⊂

+− −

+− −R R

(ii) For v Df xn( ( ¯ ))∈N , we conclude that k n v Df x v, 0k nn( ) ( ¯ )Φ = =− for all k n 1⩾ + . Using

(47), we immediately obtain v Pns( ¯ )∈R .

We apply this lemma in section 6.6 and illustrate that the given model from mathematical finance has an exponential dichotomy for one set of parameters, while the regularity condition (ii) in definition 18 is violated for a second set of parameters.

The examples that we analyze in the following sections show different characteristics con-cerning their invertibility.

Section 6.2: An invertible system with identical stable and unstable dichotomy rates.Section 6.3: An invertible three-dimensional map with a two-dimensional stable manifold.Section 6.4: A model for which Df xn( ¯ ) is noninvertible for all n Z∈ .Section 6.5: A map—showing wild chaos—that is noninvertible but still locally invertible

near each finite x 02 \ R∈ .Section 6.6: A model from mathematical finance for which Df xn( ¯ ) is noninvertible for exactly

one n Z∈ .

6.2. The invertible Hénon map

Our first example is the well-known Hénon map, which plays the role of a normal form for invertible two-dimensional quadratic maps (see [13]):

⎛

⎝⎜

⎞

⎠⎟= + −F

xx

x axbx

1 .1

22 1

2

1( ) (48)

A generalized version of the Hénon map with nontrivial orientation properties was studied in [11] and [9].

We consider here the parameters a = 1.4 and b = −1, for which the Hénon map (48) is orientation preserving and possesses two primary homoclinic orbits. As in steps (Ia) and (Ib) of the algorithm, these are determined via parameter continuation. The resulting orbits are depicted in figure 4, together with a continuation diagram. The variational equations (1) along these homoclinic orbits turn out to have exponential dichotomies with identical rates

1.58s u¯ ¯α α= ≈ .


3371

Next, we execute steps (II)–(V) of our algorithm and obtain figure 5, which reveals the local structure of the maximal invariant set by zooming into the black square of figure 4. In our diagrams we display the coding of the computed homoclinic points by a shorter sequence s k k A, 1[ ] ( )∈ Σ− + . Due to the construction in step (II) of the algorithm, the symbolic sequence s k k,[ ]− + satisfies condition (15), which allows for shorter coding, realized by colored tuples. We explain the coding by taking the example from figure 1. Take the sequence in A 12( )Σ

10 1 0 0 0 0 0 0 0 2 0 ,

where the 1 denotes the center element at position 0. We count the zero symbols on the left and right of the center element and indicate this by the coding (4, 3), 4 is colored in blue and 3 in magenta. The blue color of 4 symbolizes that the orbit on the left is of type 1, while the magenta color of 3 indicates an orbit of type 2 on the right.

Figure 5 shows the gain in information when passing over from 1= primary homoclinic orbit (left diagram) to 2= orbits (right diagram). Note that points with the (magenta-blue) coding (1, i), i ∈ …1, , 10 lie outside the plotting area of figure 5 (right).

We now consider three-humped orbits of type 1= and derive an estimate between certain points in the maximal invariant set. For N Nsol⩾ , let z1,2

Z be two homoclinic orbits, coded by (i, k) and (j, k), respectively. The left diagram in figure 5 shows a small common neighborhood of zn

1,2¯ for n 0¯ = .

Denote by s1,2˜Z the corresponding coding in A 11( )Σ and let s g s: , 11,2 1,2( ˜ )Z Z= be the coding in

A N( )ZΣ , where g is defined in (14).

Since the second symbols in the short coding coincide, we conclude that s sn n1 2= for

⩾ = − − +−n m i j: max , 1. Using theorem 8 on the interval m m,[ ]− + (which applies for suf-ficiently large m+ ), we obtain in the limit m → ∞+ for n m¯ ⩾ −

z z L e .n nn m1 2

sols∥ ∥ ⩽¯ ¯( ¯ )ε− β− − − (49)

Figure 4. Two homoclinic orbits (blue and magenta in the upper right picture) of (48), found via numerical continuation w.r.t. parameter b (left diagram). The lower right figure shows the stable (green) and unstable (red) manifolds of the fixed point ξ. A zoom-in of the black square is given in figure 5.


3372

Thus, the distance between these points essentially depends on the stable dichotomy rate. Note that this result also holds true for finite time computations, see corollary 9. Furthermore, our numerical experiments suggest that the corresponding points of these orbits lie on the same segment of the stable manifold.

If we consider the coding symbols (k, i) and (k, j) with identical first entries, then stable has to be replaced by unstable in the discussion above. The corresponding estimate from theorem 8 reads with m i j N: min , 2 = + −+ and n m¯ ⩽ +:

z z L e .n nm n1 2

solu∥ ∥ ⩽¯ ¯( ¯)ε− β− −+ (50)

6.3. A three-dimensional invertible Hénon map

We demonstrate in this section that our algorithm from section 2.2 can easily be applied to higher-dimensional systems. In contrast to the two-dimensional case, meaningful presentation of the numerical output turns out to be an additional challenge. We meet this challenge using the contour algorithm, see [20, section 4], which allows us to approximate two-dimensional invariant manifolds in three-dimensional discrete time systems.

Consider the three-dimensional version of Hénon’s map proposed in [7, example 2]:

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

+ −= =F

xxx

a bx xxx

a bwith 1.4, 0.3.1

2

3

3 12

1

2

(51)

Figure 5. Zoom-in of the black square from figure 4. Points in the maximal invariant set are computed by the algorithm from section 2.2 with J = [−3, 3], k k 10− = =− + . Humps of 1= type only (left) and of both types (right). Note that the distance between points of type (i, k), ( j, k) (resp. (k, i), (k, j)) converges to zero as k increases, see (49) and (50) resp.


3373

First, we compute a homoclinic orbit w.r.t. the fixed point ξ, where

b ba i

1

2

1

4, 1, 2, 3.i

2( )ξ =

−+

−+ =

The matrix DF( )ξ has one unstable and two stable eigenvalues and, consequently, the stable manifold is two-dimensional, while the unstable manifold is one-dimensional.

In the second step, we apply numerical continuation w.r.t. parameter b, resulting in the closed loop of homoclinic orbits shown in the left diagram in figure 6. The right diagram dis-plays the x1-components of the corresponding orbits for b = 0.3. Note that (51) implies that the other two components are time-shifted versions of the first one, i.e. the x1-component at time n equals the x2-component at time n + 1 and the x3-component at time n + 2. A phase plot of these orbits, together with stable and unstable manifold approximations, is given in figure 7.

An analysis of the local structure of the maximal invariant set inside the black box in figure 7 is carried out as described in the previous section. Figure 8 illustrates the output of this procedure.

6.4. A noninvertible Hénon example

Denote by ξ≈ −1.82, 1.82( ) a fixed point of (48) with stable and unstable eigenvalues and vectors

DF x x DF x x,s s s u u u( ) ( )ξ λ ξ λ= =

and let s u,V be the corresponding stable and unstable subspaces. We modify this system such that the stable eigenvalue is 0, while the unstable eigenvalue is unchanged. In this way, we create the noninvertible system

R( ) ( ) ( ( ) ) ( ) ( )ξ= + ⋅ − = − ⋅ ∈−G x F x A F x A x x x: , where : , 0 , .s s u1 2,2 (52)

It turns out that the dynamics are reduced to a one-dimensional line : u ξ= +D V .

Lemma 16. For each x 2R∈ , we obtain G x( ) ∈D.

Proof. For x 2R∈ let y x ξ= − . It follows that

G x G y F y A F y

I A F DF y y A

I A z z z z z z

d

,s u s u s u

0

1

( ) ( ) ( ) ( ( ) )

( ) ( ) ( )

( )( )

⎛⎝⎜

⎞⎠⎟∫

ξ ξ ξ ξ

ξ ξ τ τ ξ

ξ ξ ξ

= + = + + ⋅ + −

= + + + ⋅ −

= + + + = + + − = +

where z DF y yd0

1( )∫ ξ τ τ= + ⋅ , z zs u, s u,= |V .

As a consequence, it suffices to analyze the one-dimensional reduced system, defined as

G x x, where .u u( ) ( ) ( )λ µ λ ξ λ ξ µ λ+ = + (53)

The left diagram in figure 9 shows a homoclinic orbit as well as stable and unstable manifolds of ξ for the two-dimensional system (52). The right diagram illustrates the corresponding data for the reduced system. It turns out that the variational equation along the homoclinic orbit in the full system (52) has an exponential dichotomy, where the range of the unstable projector Pn

u satisfies Pnu u( ) =R V for all n Z∈ .


3374

Note that the fixed point 0 in the reduced system (53) is unstable and has two pre-images. Thus, homoclinic orbits may exist and in this case, the unstable fixed point is called a snap-back repeller in the literature. One of the first proofs of chaos (in the sense of Li and Yorke, see [4, defenition 3.1]) in this noninvertible context came from Marotto, see [26] and the ensuing literature.

In figure 10, we apply the algorithm from section 2.2 to compute the points in the intersec-tion of the maximal invariant set with the black box from figure 9. The zoom-in in figure 10 illustrates the fractal structure of the maximal invariant set.

For k N∈ , the set of tuples k i k i, : , :( ) ( ) N⋅ = ∈ symbolizes the coding of the corresp-onding points of the homoclinic orbits. In this example, 0sλ = and, hence, the dichotomy rate sβ can be chosen arbitrarily large. With the notation from section 6.2, equation (49) gives z z 0n n

1 2∥ ∥¯ ¯− = for orbits z1,2Z with symbolic coding (i, k) and ( j, k), respectively, and

− − +n i jmax , 1¯ ⩾ . Thus, the orbit points zn1,2¯ with the coding k,( )⋅ coincide and our numer-

ical approximations of these points are identical up to machine accuracy.The maximal invariant set contains further points in the intersection of the stable and unsta-

ble manifolds. These points belong to different primary homoclinic orbits, as described in

Figure 6. Continuation of the homoclinic orbits of (51) w.r.t. b (left). The right diagram shows two orbits for b = 0.3.

Figure 7. The two homoclinic orbits from figure 6, shown with the unstable manifold of ξ (red, both panels) and parts of the two-dimensional stable manifold (right panel). A zoom-in of the small black box in the right panel is provided in figure 8.


3375

section 2.1. There is a multitude of such orbits, since each point of a homoclinic orbit has two pre-images.

6.5. A model for wild chaos

The notion of wild chaos refers to robust homoclinic tangencies between stable and unstable manifolds of a hyperbolic set. Wild chaos can occur in vector fields, diffeomorphisms and noninvertible difference equations with at least four, three and two dimensions, respectively. The model that we analyze here originates from [5],

⎛

⎝⎜

⎞

⎠⎟λ λ

=− + − +

F

xxx

x

x

x xx x

c

: 0 ,

1

2.

a

2 2

1

2

2

22

12

22

1 2

R R

( )\ →

∥ ∥∥ ∥

(54)

The authors of [5] reduce the flow of a special vector field on the solid torus nT of dimension n 5⩾ to the dynamics of this map. Detailed investigations of wild chaos for this map can be found in [15–17, 28]. The authors use Cl_MatContM to compute homoclinic orbits and their tangencies by solving suitable boundary value problems.

Figure 8. Zoom-in of the black box from figure 7. Points in the maximal invariant set are computed via the algorithm from section 2.2 with J = [−4, 4], k k 10− = =− + . The upper diagram shows the (visible) computed points and parts of the one-dimensional unstable manifold (red) and the two-dimensional stable manifold of ξ. The lower diagram appears when the upper one is rotated slightly, such that the leaves almost collapse to lines. In the latter figure we also illustrate the coding of the homoclinic points.

x1

x1

x2

x2

x3

x3


3376

We use similar techniques but concentrate on calculating the maximal invariant set. As in [15, figure 19d], our parameters are

a c0.8, 0.8, 1.30.2

.⎜ ⎟⎛⎝

⎞⎠λ= = =

We illustrate the maximal invariant set by computing the homoclinic orbits, together with the stable and unstable manifolds, of the fixed point.

For a more global picture, we transform the system to the square [−1, 1]2 via the trans-

formation T x x

x1( )

∥ ∥=

+ ∞, see figure 11. This transformation is better suited to our contour

algorithm than the Poincaré disk used in [15, 16], and we accept the loss of smoothness caused by this transformation on the diagonals. An illustration of the maximal invariant set within the black box in figure 11 is given in figure 12.

The dichotomy rates of the variational equation along the homoclinic orbit are 0.67sβ ≈ and 0.26uβ ≈ . Comparing estimates (49) and (50), we expect—in agree-

ment with our numerical computations—that the points with coding symbols from the set (∙, k) := (i, k) : i ∈ N ∪ (i, k) : i ∈ N for example, are distinct, but lie in a small com-mon neighborhood. These points cannot be distinguished within the graphical resolution of figure 12. For details on the color coding in the case of 2= primary humps, see section 6.2 and figure 1.

6.6. A noninvertible model of asset pricing

The following system describes the mean (x1) and variance (x2) of asset prices

F x

xx

x xx

,

1

11

1

,

12

2 12

2

2( )

( )

⎛

⎝

⎜⎜⎜⎜⎜

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎞

⎠

⎟⎟⎟⎟⎟

γ

δδ

γ

δ δγ

=

+−−

+ −−

−

(55)

where the parameter 0.8δ = is fixed and the parameter γ will be varied. The underlying model, see [8, equation (23)], is based on the interaction of two groups of investors, one of which believes in the fundamental value of assets, and the other of which believes only in statisti-cal data. This model exhibits homoclinic orbits w.r.t. the fixed point 0ξ = , which proves that chaotic dynamics exist in the case where assumption (A4) is satisfied.

In the context of this paper, we are particularly interested in the noninvertible setup, in which a point of the homoclinic orbit lies on the critical set x D F x: det , 0x0

2( ) ( ( )) Rγ γ= ∈ =J ; see [27, chapter 3] for more detail on this Ligne critique.

We find this codimension-one phenomenon for (55) by applying continuation tech-niques, see [1], to homoclinic orbits in combination with a bisection scheme. At the value γ ≈ −0.027 796 780 528 093 43, the sets W ,s( ¯)ξ γ , W ,u( ¯)ξ γ and 0( ¯)γJ have a common point of intersection yn. A plot of the corresponding homoclinic orbit yZ is given in the left diagram in figure 13.

It turns out that our assumptions (A1)–(A4) are satisfied for the parameter value γ. Denote by Pn

s u, , n Z∈ the dichotomy projectors of the corresponding variational equation. Lemmas 15 (ii) and 17 give the following relations between the dichotomy projectors and the range and nullspace of D F y ,x n( ¯)¯ γ :


3377

γγ ξ γ

⊂= ⊂+ +

D F y P

D F y P T W

, ,

, , .x n n

s

x n nu

yu

1 n 1

N R

R R

( ( ¯)) ( )( ( ¯)) ( ) ( ¯)

¯ ¯

¯ ¯ ¯

Lemma 17. Assume that f satisfies (A1)–(A4) for space dimension d = 2. Further assume that Pdim 1n

s( ¯ )¯ =R and that Df xdim 1n( ( ¯ ))¯ =N .Then Df x P T Wn n

ux

u1 n 1( ( ¯ )) ( ¯ ) ( )¯ ¯ ¯ ¯⊂ ξ= + +R R .

Proof. If follows from (A4) that Df x P P:n nu

nu

1( ¯ ) ( ¯ ) → ( ¯ )¯ ¯ ¯+R R is invertible. Since Df xn( ( ¯ ))¯R and Pn

u1( ¯ )¯+R are one-dimensional subspaces, they coincide and the claim follows with

lemma 15 (i).

The maximal invariant set in a neighborhood of yn is displayed in the right panel of figure 13. The dichotomy rates of the variational equation along the homoclinic orbit are

0.22sβ ≈ and 1.86uβ ≈ . From the estimates (49) and (50), we expect—in accordance with our

Figure 9. Left diagram: homoclinic orbit of (52) (black points) with stable (green) and unstable (red) manifolds of the fixed point ξ. The right diagram shows the same homoclinic orbit in the reduced system (53) as well as its snap-back behavior.

−2 −1 0 1 20

0.5

1

1.5

2

2.5

3

0 1 2 3−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x1

x2 ξ

λ

µ(λ)

Figure 10. Maximal invariant set within the black box from figure 9. For the computation we use k k 10− = =− + and J = [−3, 4].

2.6 5.2 7.8 10.4

5

10

15

20

( ,1) ( ,2) ( ,3)

( ,10). . .

.

x1

x2

·10−6 + 1.624895

·10−6 − 0.861826


3378

numerical results—that the points with a symbolic coding from the set k k i i, : , :( ) ( ) N⋅ = ∈ are distinct but lie in a small common neighborhood. Note that we do not plot the unstable set passing through these points in this tiny neighborhood, since a fast algorithm that works accurately for this task is lacking.

Finally, we consider a second parameter value γ ≈ −0.042 126 514 636 736 42˜ for which the stable and unstable sets as well as 0( ˜)γJ also have a common point of intersection, see figure 14. Denote by zZ the corresponding homoclinic orbit and let zn be the point of this orbit, lying on 0( ˜)γJ .

Figure 11. Two homoclinic orbits of (54), together with an approximation of the stable (green) and unstable (red) sets of the fixed point ξ.

ξ

xn

Figure 12. Maximal invariant set within the black box from figure 11. For the computation we use k k 10− = =− + and J = [−10, 9], 2= .

−0.136 −0.134 −0.132 −0.13 −0.128 −0.126 −0.124 −0.122

0.505

0.51

0.515

0.52

.( , )10

( , ) ( , )

( , )( , ).

..

.4

4

3 3

x1

x2 xn


3379

But now the assumption (A4) is not satisfied, since the regularity condition (ii) in definition 18 is violated. To prove this statement, assume to the contrary that the variational equation has an exponential dichotomy with projectors Pn

s u, , n Z∈ . Note that

( ) ˜ ˜⎜ ⎟⎛⎝

⎞⎠

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟γ

δδ

γδ

=+

−−D F x x0 ,

10

0x

2 2

Figure 13. Left diagram: stable (green) and unstable (red) sets of the fixed point ξ, together with the homoclinic orbit segment yJ, J = [−40, 40] (black points). The critical curve 0( ¯)γJ is shown in blue. At the point yn 0( ¯)¯ γ∈J , the maximal invariant set in a neighborhood of yn is depicted in the right diagram for k k 10− = =− + and J = [−14, 14].

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.1 0.103 0.106

0.298

0.3

0.302

0.304

0.306

.(10, )

.(1, )

x1x1

x2 x2

ξ

yn

yn

J0(γ)

J0(γ)

R(DxF (yn, γ))

N (DxF (yn, γ))

Figure 14. Left diagram: stable (green) and unstable (red) sets of the fixed point ξ, together with the homoclinic orbit segment zJ, J = [−40, 40] (black points). The critical curve 0( ˜)γJ is shown in blue. Zoom-ins of the red box are given in the right diagrams. The upper figure illustrates the tangential intersections of the stable and unstable sets at the point zn 1˜+ . The lower diagram shows an extensive continuation of the unstable set, indicating additional transversal intersections close to zn 1˜+ , but not exactly at zn 1˜+ .

−0.1 −0.05 0 0.05 0.1 0.150

0.05

0.1

0.15

0.2

0.25

0.3

−1 −0.5 0 0.5 10.16

0.165

0.17

−1 −0.5 0 0.5 10.16

0.165

0.17

x1

x2

ξ

zn

zn+1

zn+1

J0(γ) R(DxF (zn, γ))

N (DxF (zn, γ))


3380

and this matrix is invertible for x x : 0.2072 21˜ γ≠ = + ≈δδ− . It turns out that z

x0

n2

˜ ⎜ ⎟⎛⎝

⎞⎠=

and, consequently, D F z ,x ( ˜)γ is invertible for all n 1⩾ ˜ + . On the one hand, lemma

17 yields P D F z , : 01

:nu

x ns

1 ( )( ) ( ( ˜)) ˜ ˜ Rγ λ λ= = = ∈+R R V . On the other hand, sV

is invariant w.r.t. n 1,( ˜ )Φ + for all n 1⩾ ˜ + , and we conclude with lemma 15 (ii) that P n P n D F z n1, 1, , 1,n

s sx n

s s1 ⊃( ) ( ˜ ) ( ) ( ˜ ) ( ( ˜)) ( ˜ )˜ ˜ γ= Φ + Φ + = Φ + =+ R R N V V for all

n 1⩾ ˜ + . As a consequence, P P 0ns

nu

1 1( ) ( ) ˜ ˜∩ ≠+ +R R , and thus the variational equation can-not have an exponential dichotomy. In particular, we see for this parameter setup that our theorems do not apply.

Note that the simultaneous occurrence of transversal and tangential intersections of sta-ble and unstable sets along the same homoclinic orbit is not generic; see the discussion in [35, section 4].

Acknowledgments

The authors are grateful to both referees for their detailed comments, which improved this paper.

This work was supported by CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’.

Appendix A. Exponential dichotomy

In this appendix, we summarize important results from the theory of exponential dichotomies in a noninvertible setup.

Let J n n,[ ]= − + , ˜ [ ]= −− +J n n, 1 , where the cases = −∞−n and n = ∞+ are permitted. Consider the nonautonomous linear difference equation

u A u A n J, ,n n n nd d

1, ˜R= ∈ ∈+ (A.1)

and denote by m n A A m n, ,m n1( ) ⩾Φ = − its forward solution operator. We define the notion of an exponential dichotomy, see [14, definition 7.6.1], that is called regular in [2, definition 3.1] and [22, definition 2.1.2].

Definition 18. The difference equation (A.1) has an exponential dichotomy on J with data K P, ,s u n

s u,

,( )α , if there exist constants K , , 0s uα α > and families of projectors Pns, P I P:n

uns= − ,

n J∈ such that

(i) P n m n m P, ,ns u

ms u, ,( ) ( )Φ = Φ for all n m J, ∈ , n m⩾ ;

(ii) A P P:n P nu

nu

1nu ( ) → ( )( )| +R RR is invertible for all n J∈ with the inverse of n m, Pm

u( ) ( )Φ |R , n m⩾ denoted by m n P P, : n

umu¯ ( ) ( ) → ( )Φ R R ; and

(iii) for n m J, ∈ , n m⩾ the following estimates hold:

n m P K m n P K, e , , e .ms n m

nu n ms u∥ ( ) ∥ ⩽ ∥ ¯ ( ) ∥ ⩽( ) ( )Φ Φα α− − − −

Denote by

x x: :J Jd J

J ( ) ∥ ∥ R= ∈ < ∞∞S

the Banach space of bounded sequences indexed by J. In the case J Z= , the regularity condi-tion (ii) in definition (18) implies that the inhomogeneous equation


3381

u A u r n r, ,n n n n1 Z Z Z= + ∈ ∈+ S (A.2)

has a unique bounded solution in Z, see [14, theorem 7.6.5], [2, theorem 4.5].

Lemma 19. Assume that the difference equation (A.1) has an exponential dichotomy on Z with data K P, ,s u n

s u,

,( )α . Then

u G n i r n G n mn m P n m

n m P n m: , 1 , , , :

, , for ,

, , forn

ii

ms

mu¯ ( ) ( )

( ) ⩾¯ ( )

ZZ

⎧⎨⎩∑= + ∈ =

Φ−Φ <∈

(A.3)

is the unique bounded solution of (A.2) on Z.

Note that in the case J n n,[ ]= − + with finite n− or n+ , extra boundary conditions are needed to guarantee the uniqueness of the solution uJ.

Finally, we cite the most important perturbation result for exponential dichotomies, the roughness theorem, see [2, theorem 4.4], [22, Satz 3.2.1] and [14, theorem 7.6.7].

Theorem 20. Assume that the difference equation (A.1) has an exponential dichotomy on J with data K P, ,s u n

s u,

,( )α . Let 0 s u s u, ,β α< < . Then there exist positive constants K , ,s u s u, ,( )γ γ α β= and K , , ,s u s u, ,( )µ µ α β= , such that for any sequence of matrices BJ

d d J,( )R∈ , satisfying BJ∥ ∥ ⩽ γ∞ , the perturbed equation

u A B u n J,n n n n1 ( ) ˜= + ∈+ (A.4)

has an exponential dichotomy on J with data K Q2 , ,s u ns u

,,( )β , where

P Q K B n J2 for all .ns

ns

J∥ ∥ ⩽ ∥ ∥ µ− ∈∞ (A.5)

Note that explicit formulas for γ and μ may be derived from [22, section 3], and that these constants do not depend on J.

Sketch of proof. The proof is a noninvertible version of the proof by Kleinkauf [24, lemma 1.1.9], [23, lemma 2.3], who followed an idea of Sandstede’s [36, lemma 1.1] for continuous systems. For details, the reader is referred to [38, Satz 2.1]. For 0,s u s u, ,( )β α∈ , introduce the exponentially weighted Banach space

= ∈ < ∞×∗X X:J

d d J J,Z R ( ) ∥ ∥

with the norm X X n m X n mmax sup e , , sup e ,n mn m

n mm ns u∥ ∥ ∥ ( )∥ ∥ ( )∥⩾

( ) ( )= β β∗

−<

− . Consider the operators T T, : J J

d d d d J J1

, ,( ) → ( )R R× ×Z S defined by

T X B n m G n l B X l m n m J J

T X B G T X B

, , , 1 , , , ,

, , ,

Jl J

l

J J

1

1

( )( ) ( ) ( ) ( )

( ) ( )

∑= + ∈ ×

= +∈

where G is the Green function of the unperturbed system from (A.3). A computation shows that T1 is uniformly Lipschitz continuous in the second argument:

T B B, J J1∥ ( )∥ ⩽ ∥ ∥µ⋅ ∗ ∞

with ,s u s u, ,( )µ µ α β= independent of BJ. A crucial step is to show that X J∈Z is a fixed point of T B, J( )⋅ if and only if X solves the equation

X n m A B X n m I n J m J1, , , ,n n n m, 1( ) ( ) ( ) ˜δ+ = + + ∈ ∈− (A.6)

subject to the boundary conditions


3382

( ) ( )

= −∞ <= < ∞

− −

+ +

−

+

P X n m n

P X n m n

, 0, if ,

, 0, if ,ns

nu (A.7)

respectively. Here, δ denotes the Kronecker symbol.To prove this equivalence, we observe that if X is a fixed point of T B, J( )⋅ , then equa-

tions (A.6) and (A.7) follow from a direct computation. For the proof of the converse state-ment, one studies the inhomogeneous linear equation for fixed m,

Y n m A Y n m B X n m I1, , , ,n n n m, 1( ) ( ) ( ) δ+ − = + −

which possesses the solution Y = X by (A.6). Uniqueness is proven by using the dichotomy properties to show that the associated homogeneous equation only possesses the trivial solu-tion. An application of a matrix-valued analog of lemma 19 then yields the fixed point property.

The existence of a fixed point, and thus of a solution of (A.6), follows with the Banach fixed point theorem. Since T B, J( )⋅ is a uniform contraction on JZ w.r.t. the norm ∥ ∥⋅ ∗ for suf-ficiently small BJ∥ ∥∞, we obtain the existence and uniqueness of a fixed point X BJ¯ ( ) with a smooth dependence on BJ. Using the uniqueness, one shows that Q X B n n,n

sJ¯ ( )( )= is a pro-

jector and that the perturbed equation (A.4) possesses an exponential dichotomy with data K Q2 , ,s u n

s u,

,( )β . Finally, one proves that X BJ¯ ( ) is the Green function for this system.

Appendix B. A Lipschitz inverse mapping theorem

The proof of theorem 7 uses a quantitative version of the Lipschitz inverse mapping theorem, see [21, appendix C].

Theorem 21. Assume that Y and Z are Banach spaces, F C ,1( )∈ Y Z and let F y0( )′ be a homeomorphism for some y0 ∈Y. Let , , 0 κ η δ> be three constants, such that the following estimates hold:

F y F yF y

y y1

,00

1 0( ) ( ) ⩽ ⩽( )

( )κ η− < ∀ ∈′ ′′

δ−B (B.1)

F y .0( ) ⩽ ( )η κ δ− (B.2)

Then F has a unique zero y y0¯ ( )∈ δB .

References

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