stability in nonautonomous dynamics - barreira

8/12/2019 Stability in Nonautonomous Dynamics - Barreira

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STABILITY IN NONAUTONOMOUS DYNAMICS:

A SURVEY OF RECENT RESULTS

LUIS BARREIRA AND CLAUDIA VALLS

Abstract. We discuss recent results in the stability theory of nonau-tonomous differential equations under sufficiently small perturbations.We mostly concentrate on the description of our own work, with empha-sis on the Lyapunov stability of solutions, the existence and smoothnessof invariant manifolds, and the construction and regularity of topologicalconjugacies, among other topics. The main novelty is that we alwaysconsider a nonuniform exponential b ehavior of the linear variationalequations, given either by the existence of a nonuniform exponentialcontraction or a nonuniform exponential dichotomy.

Contents

1. Introduction 12. Exponential contractions and stability 73. Exponential dichotomies 114. Stable invariant manifolds 145. Construction of conjugacies 226. Center manifolds and reversibility 257. Lyapunov regularity 28References 31

1. Introduction

The unifying theme of this survey is the stability of nonautonomous dif-ferential equations under sufficiently small nonlinear p erturbations, withemphasis on the study of the Lyapunov stability of solutions, and of theexistence and smoothness of invariant manifolds. We describe recent resultsin the area, mostly concentrating on our own work. The main novelty isthat we always consider a nonuniform exponential behavior of the linearvariational equations. This causes that the stability results discussed inthe survey hold for a much larger class of differential equations than in theclassical theory of exponential dichotomies. In addition, whenever possi-

ble, we consider the infinite-dimensional case. In view of the readability ofthe survey, instead of always presenting the most general results, we made

2000 Mathematics Subject Classification. Primary: 34Dxx, 37Dxx.Key words and phrases. exponential dichotomies, invariant manifolds, stability theory.Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Sys-

tems, and through Fundacao para a Ciencia e a Tecnologia by Program POCTI/FEDER,Program POSI, and the grant SFRH/BPD/26465/2006.

1


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2 LUIS BARREIRA AND CLAUDIA VALLS

an appropriate selection of the material, although this selection should alsoreflect a personal taste.

1.1. Exponential dichotomies and generalizations. In the theory ofdifferential equations, both in finite-dimensional and infinite-dimensionalspaces, the notion of exponential dichotomy, introduced by Perron in [57],plays a central role in the study of the stability of solutions, and in particularin the theory of stable and unstable invariant manifolds. For example, letu(t) be a solution of the equation u =F(u), for some differentiable map Fin a Banach space. Setting A(t) = du(t)F, the existence of an exponentialdichotomy for the linear variational equation

v =A(t)v, (1)

with some additional assumptions on the nonlinear part of the vector field,guarantees the existence of stable and unstable invariant manifolds for thesolution u(t). We note also that the theory of exponential dichotomies andits applications are quite developed. In particular, there exist several classesof linear differential equations with exponential dichotomies. We can men-tion, for example, the series of papers of Sacker and Sell [71, 72, 73, 70, 74]that in particular discuss sufficient conditions for the existence of exponentialdichotomies, also in the infinite-dimensional setting. In a different direction,geodesic flows on compact smooth Riemannian manifolds with negative sec-tional curvature have the whole unit tangent bundle as a hyperbolic set, i.e.,they define Anosov flows. For any such flow the linear variational equationof each trajectory has an exponential dichotomy. Furthermore, time changesand small C1 perturbations of flows with a hyperbolic set also possess hy-perbolic sets (see for example [41]). We refer to the books [21, 35, 38, 75]for details and further references related to exponential dichotomies. Weparticularly recommend [21] for historical aspects. One may also consultthe books [27, 28, 50], in spite of being less recent. On the other hand, even

though all the literature and the intense research activity in the area overthe years, the notion of exponential dichotomy considerably restricts thedynamics. It is thus important to look for more general types of hyperbolicbehavior.

Our main objective is precisely to consider the more general notion ofnonuniform exponential dichotomyand study in a systematic manner someof its consequences, in particular in connection with the existence and thesmoothness of invariant manifolds for nonautonomous differential equations.We also discuss a version of the GrobmanHartman theorem, the existenceof center manifolds, as well as their reversibility properties, and Lyapunovsregularity theory and its applications to the stability of solutions of nonau-tonomous equations. In comparison with the classical notion of (uniform)

exponential dichotomy, the existence of a nonuniform exponential dichotomyis a much weaker assumption. In particular, p erhaps surprisingly, we willsee that in a certain sense any linear equation as in (1), having global so-lutions and at least one nonzero Lyapunov exponent, admits a nonuniformexponential dichotomy.

1.2. Nonuniform hyperbolicity. Our work is also a contribution to thenonuniform hyperbolicity theory, with emphasis on differential equations


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STABILITY IN NONAUTONOMOUS DYNAMICS 3

and with a systematic development in the infinite-dimensional setting. Werefer to [1, 3] for detailed expositions of parts of the nonuniform hyperbolic-ity theory and [2] for a detailed description of its contemporary status. Thetheory goes back to the landmark works of Oseledets [54] and particularlyPesin [58, 59, 60]. Since then it became an important part of the generaltheory of dynamical systems and a principal tool in the study of stochas-

tic behavior. We note that the nonuniform hyperbolicity conditions can beexpressed in terms of the Lyapunov exponents. For example, almost all tra-jectories of a dynamical system preserving a finite invariant measure withnonzero Lyapunov exponents are nonuniformly hyperbolic. Among the mostimportant properties due to nonuniform hyperbolicity is the existence of sta-ble and unstable invariant manifolds, and their absolute continuity propertyestablished by Pesin in [58]. The theory also describes the ergodic proper-ties of dynamical systems with a finite invariant measure that is absolutelycontinuous with respect to the volume [59], and expresses the KolmogorovSinai entropy in terms of the Lyapunov exponents by the Pesin entropyformula [59] (see also [45]). In another direction, combining the nonuniformhyperbolicity with the nontrivial recurrence guaranteed by the existence of

a finite invariant measure, the fundamental work of Katok [40] revealed avery rich and complicated orbit structure, including an exponential growthrate for the number of periodic points measured by the topological entropy,and the approximation of the entropy of an invariant measure by uniformlyhyperbolic horseshoes (see also [3]).

We now consider the particular case of the stable manifold theorem, andwe compare the existing results with our own work. We first briefly describethe relevant references. The proof by Pesin in [58] is an elaboration of theclassical work of Perron. In [68] Ruelle obtained a proof of the stable mani-fold theorem based on the study of perturbations of products of matrices inOseledets multiplicative ergodic theorem [54]. Another proof is due to Pughand Shub in [66] with an elaboration of the classical work of Hadamard us-

ing graph transform techniques. In [31] Fathi, Herman and Yoccoz provideda detailed exposition of the stable manifold theorem essentially followingthe approaches of Pesin and Ruelle. There are also versions of the stablemanifold theorem for dynamical systems in infinite-dimensional spaces. In[69] Ruelle established a version in Hilbert spaces, following his approachin [68]. In [48] Mane considered transformations in Banach spaces undercertain compactness and invertibility assumptions, including the case of dif-ferentiable maps with compact derivative at each point. On the other hand,in [65] Pugh constructed a C1 diffeomorphism in a manifold of dimension 4,that is not of class C1+ for any , and for which there exists no invariantmanifold tangent to a certain stable space such that the trajectories alongthe invariant manifold travel with exponential speed. But this example doesnot forbid the existence of a C1 dynamics, which is not of class C1+ forany , for which there still exist stable manifolds. We show in [5, 7] thatthere exist stable invariant manifolds for the nonuniformly hyperbolic trajec-tories of a large family of dynamics that, in general, are at most of class C1.Furthermore, in [9] we obtain Ck stable manifolds in Banach spaces, usinga single fixed point problem to obtain all derivatives simultaneously.


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1.3. Uniform contractions and nonuniform contractions. In order todescribe the main problems in which we are interested, we first introduce thenotions of uniform contraction and nonuniform contraction. We note thatthese are simpler notions than those of (uniform) exponential dichotomy andnonuniform exponential dichotomy. On the other hand, this simplificationallows us to describe the main ideas without accessory technicalities.

Consider the linear equation (1) in a Banach space X, for a family ofbounded linear operators A(t) varying continuously with t 0. We assumethat (1) has unique solutions that are defined for all time t 0. We alsoconsider the equation

v =A(t)v+ f(t, v) (2)

for a continuous function f such thatf(t, 0) = 0 for every t 0. In partic-ular, v(t) = 0 is still a solution of equation (2).

When there exist constants c >0 and 0, where

a(t) = sin log t + cos log t.

For example, for the perturbation f(t, (x, y)) = (0, x4), one can show thatalthough the limsup in (4) is constant and equal to 1, there exist solutions

v(t) of equation (2) for whichlimsupt+

1

tlogv(t)> 0

(we refer to [1] for details). In other words, assuming that all Lyapunovexponents are negative is not sufficient to guarantee that the asymptoticstability of the zero solution of equation (1) persists under sufficiently smallnonlinear perturbations.


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We now briefly explain the difficulty in establishing the asymptotic stabil-ity in equation (2) when the ambient spaceX is finite-dimensional. Assumethat condition (4) holds, i.e., that all Lyapunov exponents are negative.Since the space of solutions of equation (1) is finite-dimensional (since it hasthe same dimension as X), one can easily show that there exist constantsD >0, 0 the estimate in (5) can be shown to imply that thesize of the neighborhood at time s where the exponential stability of thezero solution of equation (2) is guaranteed decreases with exponential rateat least (see Section 2.2). This is due to the fact that when we perturbequation (1) the exponential term es in (5) may amplify the perturbation,and thus to make sure that this is not the case we must take the initialcondition v(s) so that esv(s) is sufficiently small (that is, v(s) must beapproximately of size es). We refer to Section 2.2 for rigorous statements.

1.4. Formulation of the main problems. The former discussion alsomotivates some of the main problems that we discuss in the survey. We nowbriefly describe these problems.

Persistence of asymptotic stability. The study of asymptotic stability is atthe center of our discussion. Thus, one of the main problems is to find condi-tions that guarantee the persistence of asymptotic stability in equation (2),assuming that equation (1) admits a nonuniform exponential contraction.We describe sufficient conditions that are much weaker than the uniformasymptotic stability exhibited by a uniform exponential contraction. Fur-thermore, we also consider the situation when the constant in (5) is positiveand cannotbe made arbitrarily small.

Exponential dichotomies. We also want to consider the case when the so-lutions of equation (1) exhibit not only contraction but also expansion. Inthis case the notion of stability is replaced by the asymptotic stability alongan invariant manifold, that is tangent to the space containing the solutionsof (1) with negative Lyapunov exponent. In particular, we describe theconstruction of stable (and unstable) invariant manifolds, in the infinite-dimensional setting. We also study their regularity, with a single fixed pointproblem to obtain all derivatives simultaneously.

Robustness of nonuniform exponential behavior. Besides the problem of per-sistence of asymptotic stability in equation (1) under nonlinear perturba-

tions it is also important to know whether the stability persists under suffi-ciently smalllinear perturbations. This is the so-called robustness problem.It corresponds to consider the linear equation

v = [A(t) + B(t)]v (6)

and ask whether it admits a nonuniform exponential contraction (or a non-uniform exponential dichotomy), for a sufficiently small perturbation B,when the same happens with the original equation (1). We give conditions


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for the robustness of the nonuniform exponential behavior. We also discusshow the constants and in the notion of exponential contraction vary withthe perturbation.

Estimates for the constants determining the hyperbolicity. In the study ofthe persistence of asymptotic stability, the conditions determining the al-

lowed perturbations are expressed in terms of the constants and. Thus,it is important to obtain sharp estimates for these constants, preferablyin terms of the operators A(t), without the need to solve explicitly equa-tion (1). We describe estimates for a large class of equations. We recallthat there are examples, even within Floquet theory, with al lmatrices A(t)having only eigenvalues with negative real part, but for which there existsa positive Lyapunov exponent. This shows once more that the study of theasymptotic behavior of solutions of equation (1) is very delicate.

Construction of topological conjugacies. A fundamental problem in the studyof the local behavior of a dynamical system is whether the linearization ofthe dynamics along a given solution approximates well the solution. In other

words, we look for a local change of variables, called a conjugacy, that takesthe system to a linear one. Moreover, to distinguish the dynamics in a neigh-borhood of the solution further than in the topological category, the changeof variables must be as regular as possible. We discuss this problem in thegeneral setting of nonuniform hyperbolicity, with a version of the GrobmanHartman theorem. In addition we show that the (topological) conjugaciesare always Holder continuous.

Central behavior. It is also important to allow central directions, in whichthere is neither contraction nor expansion, at least with exponential speed,and obtain corresponding results concerning the persistence of the centralbehavior in equation (1) under sufficiently small nonlinear perturbations.

This corresponds to establish the existence of invariant center manifolds forequation (2). As is well known, these are powerful tools in the analysis of theasymptotic behavior of dynamical systems. For example, when equation (1)possesses no unstable directions, the stability in equation (2) is determinedby the behavior on (any) center manifold.

Speeds of nonuniform exponential behavior and optimal estimates. Thereare many examples of contractions for which the constant in (5) cannotbe taken equal to zero. On the other hand, in many of these examples onecan replace the term Des in (5) by a function D(s) with subexponentialgrowth, i.e., such that

lim sups+

1s

log D(s) = 0.

In this case, some of the results presented in the survey can be improved. Forexample, it is possible to obtain optimal estimates for the decay of solutionsof the perturbed equation (2) expressed in terms ofD(s) and of the constant in (5). This is a more technical problem, although the required changesare not tremendous. We refer to [16] for full details.


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2. Exponential contractions and stability

This section is dedicated to the study of the persistence of asymptoticstability of a nonuniform exponential contraction under a sufficiently smallnonlinear perturbation. Instead of starting the discussion with the more gen-eral notion of nonuniform exponential dichotomy, we start with the notionof contraction which allows us to describe the main ideas without accessorytechnicalities.

We first give nontrivial examples of nonuniform contractions, in the sensethat they are not uniform, exhibiting different speeds of nonuniform behav-ior. We then discuss the p ersistence of asymptotic stability of nonuniformexponential contractions under perturbations. We also give sharp estimatesfor the constants in the notion of nonuniform contraction. Finally, we discussthe existence of nonuniform contractions at large, in particular in connectionwith ergodic theory.

2.1. Nonuniform exponential contractions. Let B(X) be the set ofbounded linear operators in the Banach spaceX. We consider a continuousfunction A : R+

0

B(X), and the initial value problem

v =A(t)v, v(s) =vs, (7)

with s 0 and vs X. As in Section 1.3, we always assume that

each solution of (7) is defined for every t 0. (8)

We denote by T(t, s) the linear operator such that T(t, s)v(s) = v(t) foreach solution v of equation (7) and each t, s 0.

Definition 2.1. We say that equation (7) admits a nonuniform exponentialcontractionif there exist constants D >0, 0, the equation in Rgiven by

v =

+ a

t + 1(cos t 1) a log(t+ 1) sin t

v (10)

(fort >1) admits a nonuniform exponential contraction. Indeed, we have

T(t, s) =e(ts)+a log(t+1)(cost1)a log(s+1)(cos s1)

(11)(that is, the operatorT(t, s) coincides with multiplication by the exponentialin (11)), which implies that

T(t, s) (s + 1)2ae(ts), t s 0. (12)

Therefore, for each >0 there exists D >0 such that inequality (9) holds.In particular, equation (10) admits a nonuniform exponential contraction.


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We note that one is not able to take = 0 in (9): for example, takingt= 2k and s = (2l 1) with k, l Nwe find that

T(t, s) = (s + 1)2ae(ts),

and thus the estimate in (12) cannot be improved.

We now give an example of nonuniform exponential contraction for whichthe exponential termes in (9) cannot be improved.

Example 2.3 ([12, Proposition 3]). For < a < 0, the equation in Rgiven by v = (+at sin t)v admits a nonuniform exponential contraction.We have

T(t, s) =e(ts)at cos t+a sin t+as cos sa sin s

=e(a)(ts)at(cost1)+as(coss1)+a(sin tsin s).

It follows that

T(t, s) e2|a|e(a)(ts)+2|a|s, t s 0, (13)

and (9) holds with D = e2|a|

and = 2|a| > 0. Furthermore, for t = 2kand s= (2l 1) with k, l Nwe have

T(t, s) =e(a)(ts)+2|a|s ,

and thus the term 2|a|s in (13) is as small as possible.

2.2. Stability of nonuniform exponential contractions. We addressin this section the problem of the persistence of asymptotic stability, undersufficiently small perturbations of equation (7).

We consider perturbations of equation (7) given by a continuous functionf: R+0 XX such that

f(t, 0) = 0 for every t 0, (14)

and we assume that there exist constants c >0 and q >0 such that

f(t, u) f(t, v) cu v(uq + vq) (15)

for every t 0 and u, v X. We consider the initial value problem

v =A(t)v+ f(t, v), v(s) =vs, (16)

withs 0 andvs X. We note that by condition (14) the functionv(t) = 0is still a solution of equation (16).

The following result establishes the persistence of asymptotic stability ofnonuniform exponential contractions under sufficiently small perturbations.We denote by B () Xthe open ball of radius >0 centered at zero, and

we set = (1 + 1/q).Theorem 2.4. If equation(7)admits a nonuniform exponential contractionwithq + < 0, then for eachK > D there exists >0 such that for everys 0, vs, vs B(e

s) andt s we have

v(t) v(t) K e(ts)+svs vs,

wherev(t) andv(t) are the solutions of (16) withv(s) =vs andv(s) = vs.


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Theorem 2.4 is a particular case of Theorem 1 in [16]. Notice that weneed to take the initial conditionsvs and vs in a neighborhood of sizee

s.Essentially, this ensures that the exponential termes in (9) is not amplifiedin equation (16).

When = 0 we recover the following classical result.

Theorem 2.5. If equation (7) admits a uniform exponential contraction,

then for eachK > D there exists >0 such that for everys 0, vs, vs B() and t s we have

v(t) v(t) K e(ts)vs vs.

The main differences between Theorems 2.4 and 2.5 are that in the sec-ond theorem the condition q + < 0 is automatically satisfied, and thatthe neighborhoods B (es) are replaced by the single neighborhood B (),which is independent ofs.

2.3. Existence of nonuniform exponential contractions. It should beemphasized that the existence of a uniform exponential contraction (see (3)),is a very restrictive assumption. On the other hand, the existence of a

nonuniform exponential contraction (see (9)) is a more typical situation,although not only because it is a weaker assumption. To make this statementmore precise we formulate two theorems which indicate that nonuniformexponential contractions are very common, at least in finite-dimensionalspaces. We emphasize that Sections 2.3 and 2.4 consider only the finite-dimensional case. This is due to the fact that the theory is not yet developedfor arbitrary Banach spaces. We refer to [6] for an appropriate generalizationin Hilbert spaces.

The fact that condition (4) implies condition (9) can be reformulated inthe following manner.

Theorem 2.6([12, Theorem 1]). LetA(t)ben nmatrices varying contin-uously witht. If condition (4)holds then equation (1)admits a nonuniformexponential contraction.

In particular, Theorem 2.6 indicates that the constants and in (9)occur naturally in the study of asymptotic stability. Furthermore, the the-orem shows that in a certain sense the notion of nonuniform exponentialcontraction is the weakest possible notion of exponential contraction. Thus,one can argue that the approach presented in the survey is as general aspossible, although sometimes with weaker statements.

On the other hand, in view of the condition q+ 0.


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can be treated by first making a reduction to the triangular case (see [1,Lemma 1.3.3]). We refer to [12] for details.

3. Exponential dichotomies

We now consider the more general case of nonuniform exponential di-

chotomies. This section is dedicated to the linear theory; we refer to Sec-tions 46 for the discussion of the nonlinear theory, with emphasis on theconstruction of invariant manifolds and topological conjugacies.

In particular we discuss the robustness of nonuniform exponential di-chotomies, that is, whether a sufficiently small linearperturbation of a lin-ear equation with a nonuniform exponential dichotomy stills admits such adichotomy.

3.1. Nonuniform exponential dichotomies. We continue to consider acontinuous functionA : R+0 B(X), and the initial value problem (7) withs 0 and vs X. We assume that (8) holds, and we denote by T(t, s) thelinear operator such thatT(t, s)v(s) =v(t) for each solutionvof equation (7)

and each t, s 0.

Definition 3.1. We say that equation (7) admits a nonuniform exponentialdichotomy if there exists a function P: R+0 B(X) such that P(t) is aprojection for each t 0, with

P(t)T(t, s) =T(t, s)P(s) for every t, s 0, (19)

and there exist constants

D >0, a < 0 b, and 0 (20)

such that for every t s 0 we have

T(t, s)P(s) Dea(ts)+s, T(t, s)1Q(t) Deb(ts)+t, (21)

where Q(t) = Id P(t) is the complementary projection for each t 0.

The notion of nonuniform exponential dichotomy mimics the classical no-tion of (uniform) exponential dichotomy. Recall that equation (7) admitsa (uniform) exponential dichotomy if there exist a function Pas in Defini-tion 3.1 and constants D, >0 such that for every t s 0 we have

T(t, s)P(s) De(ts), T(t, s)1Q(s) De(ts).

We note that we allow the constant b in (20) to be zero. We emphasizethat this is not an essential feature of the notion of nonuniform exponen-tial dichotomy. Incidentally, in the theory of stable invariant manifolds (seeSection 4) the inequalities a


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Definition 3.2. We say that equation (7) admits a strong nonuniform ex-ponential dichotomy if there exists a function P: R+0 B(X) such thatP(t) is a projection for each t 0 and (19) holds, and there exist constants

D 0, a a n, fort= 2m and s = (2n+ 1) we have

u(t) =e(a)(ts)asu(s).

This shows that the first equation in (25) does not admit a uniform expo-nential contraction. Analogous arguments apply to the second equation.


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When equation (7) admits a nonuniform exponential dichotomy, we con-sider the linear subspaces

E(t) =P(t)X and F(t) =Q(t)X

for eacht 0. These are calledstableandunstablesubspaces att (althoughstrictly speaking F(t) should be called unstable space only when b > 0).

Clearly, X=E(t) F(t) for each t0, and the dimensions dim E(t) anddim F(t) are independent oft. Set

(t) = inf{x y: x E(t), y F(t), x= y= 1} (26)

(we took this notion from [39]). One can easily verify that in the case ofHilbert spaces we have

(t) = 2sin((t)/2),

where(t) is the angle between the subspaces E(t) and F(t). Furthermore,one can show that

1

P(t) (t)

2

P(t) and

1

Q(t)(t)

2

Q(t)

for every t 0. Setting t= s in (21) we obtainP(t) Det and Q(t) Det. (27)

Therefore, by (27),

(t) et/D for every t 0. (28)

This shows, at least in the case of Hilbert spaces, that when there existsa nonuniform exponential dichotomy the angle between the stable and un-stable subspaces cannot decrease more than exponentially, with maximalexponential rate given by the number in (20). In the general case of Ba-nach spaces, (t) in (26) can also be thought of as an angle between thespacesE(t) andF(t). We mote that in the particular case of uniform expo-

nential dichotomies we can take = 0 in (27), and thus also in (28). Thismeans that in the uniform case the angle b etween the stable and unstablesubspaces is uniformly bounded from below for all time.

3.2. Robustness of nonuniform exponential dichotomies. We con-sider continuous functions A, B : R+0 B(X), and equation (6). Set =min{a, b}. For a given constant >0 we define

=

1 2D/ and D= 4D2

1 D/(+ ). (29)

Then for every t s 0 we have

T(t, s)P(s) De(ts)+s, T(t, s)1Q(t) De(ts)+t. (30)

The following result establishes the robustness of nonuniform exponentialdichotomies for sufficiently small perturbations B .

Theorem 3.4 ([18, Theorem 2]). Assume that:

1. equation (7) admits a nonuniform exponential dichotomy withb >0and < ;

2. B(t) e2t for everyt 0.


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For each sufficiently small, equation (6)admits a nonuniform exponentialdichotomy with the constants D, , and in (30) replaced respectively by

D, , and2.

We note that by (29) the constants D and vary continuously with .Some arguments in the proof of Theorem 3.4 are inspired in work of

Popescu in [63] in the case of uniform exponential dichotomies. Inciden-tally, we note that he uses the notion usually called admissibility, while wedo not need it, of course independently of its possible interest in other sit-uations. This notion goes back to Perron and refers to the characterizationof the existence of an exponential dichotomy in terms of the existence anduniqueness of bounded solutions of the equation v =A(t)v+ g(t) for g in acertain class of bounded perturbations.

In the case of uniform exponential dichotomies the study of the robustnessproblem has a long history, in particular with work of Massera and Sch affer[49] (building on earlier work of Perron [57]; see also [50]), Coppel [26],and Dalecki and Kren [28] (in the case of Banach spaces), with differentapproaches and successive generalizations. For more recent work we refer to

[22, 53, 63, 62] and the references therein. In particular, Chow and Leiva[22] and Pliss and Sell [62] considered the context of linear skew-productflows and gave examples of applications in infinite dimension, including toparabolic partial differential equations and functional differential equations.We emphasize that in all these works it is only considered the case of uniformexponential dichotomies.

4. Stable invariant manifolds

This section is dedicated to the construction of stable invariant manifoldsfor sufficiently small perturbations of linear differential equations admittinga nonuniform exponential dichotomy. In other words, we describe versionsof the stability results in Section 2.2 when not all directions are contracting.The existence of stable invariant manifolds corresponds precisely to the ex-istence of an asymptotic stability along a certain invariant manifold, that istangent to the stable space.

4.1. Lipschitz stable invariant manifolds. Before discussing the exis-tence of smooth stable manifolds, we consider the simpler problem of ex-istence of Lipschitz stable invariant manifolds, i.e., graphs of Lipschitzfunctions along which we have asymptotic stability.

Let againXbe a Banach space, and let A : R+0 B(X) andf: R+0X

X be continuous functions such that (14) and (15) hold. In particular,v(t) = 0 is still a solution of the equation v =A(t)v+ f(t, v).

One can easily verify that when f is differentiable condition (15) is equiv-alent to the existence of constants c >0 and q >0 such thatfv (t, v)

cvq for every t 0 and v X.Condition (15) may sometimes be obtained with an appropriate cut-off ofthe perturbation in a neighborhood of 0 Xand taking a Taylor expansion,provided that the perturbation is sufficiently regular. We note that since


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all norms in R2 are equivalent, when q > 1 the q-norm (uq + vq)1/q isequivalent to the 1-normu + v. In this case, one can replace the factoruq + vq in (15) by (u + v)q , up to a multiplicative constant.

Given s 0 and vs Xwe denote by

(x(t), y(t)) = (x(t,s,vs), y(t,s,vs)) E(t) F(t)

the unique solution of the initial value problem in (16), i.e., of the problem

x(t) =T(t, s)P(s)vs+

ts

T(t, r)P(r)f(r, x(r), y(r)) dr,

y(t) =T(t, s)Q(s)vs+

ts

T(t, r)Q(r)f(r, x(r), y(r)) dr

for t s. The flow generated by the autonomous equation

t = 1, v =A(t)v+ f(t, v) (31)

is given by

(s, vs) = (s + , x(s + ,s ,vs), y(s + ,s ,vs)), 0. (32)

The stable manifolds are obtained as graphs of Lipschitz functions. Wefirst describe an appropriate class of functions. Let

= (1 + 2/q), (33)

with as in (20). We fix >0 and we consider the set of initial conditions

X= {(s, ) :s 0 and Bs(es)}, (34)

where Bs() E(s) is the open ball of radius > 0 centered at zero. Wedenote by X the space of continuous functions : X X such that foreach s 0 and , Bs(e

s) we have

(s, 0) = 0, (s, Bs(es)) F(s),

and(s, ) (s,) .

Given a function X we consider its graph

V ={(s ,,(s, )) : (s, ) X} R+0 X. (35)

We note that (s, 0) V (since (s, 0) = 0), for every s 0. In particular,V contains the line R+0 {0} and the graph

Vs= {(s ,,(s, )) : Bs(es)}

for each s 0 (see Figure 1). We also set

= + = (2 + 2/q), (36)

and we consider the corresponding sets X and X , obtained replacing by in the definitions. We will see that there exists a (unique) function X such that for every (s, ) X X the solution of (16) withvs = (, (s, )) is contained in V. This means that for this function theset V is forward invariant under the flow in (32).

The following result establishes the existence of Lipschitz stable mani-folds.


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s

s +

E

F

(s ,,(s, ))

V

Vs

Vs+

p

Figure 1. A stable manifold V of the origin. In orderthat V is invariant under the flow we require that p =(s ,,(s, )). Here the subspaces E=E(t) andF =F(t)are assumed to be independent oft.

Theorem 4.1 ([10, Theorem 3]). If equation (7) admits a nonuniform ex-ponential dichotomy with

a + 0 and a unique function X such that

(s ,,(s, )) V for every(s, ) X and0.

Furthermore, there existsK >0 such that for everys 0,, Bs(es),

and0 we have

(s ,,(s, )) (s,, (s,)) K ea+s . (38)

We call to the set V in (35) a Lipschitz stable manifoldof the origin forequation (16). In particular, setting = 0 in (38) we find that any solutionof the initial value problem in (16) starting in V, i.e., with vs = (, (s, ))for some Bs(e

s), approaches zero with exponential speed a (whichis independent of s and ). We note that both inequalities in (37) are

automatically satisfied when is sufficiently small, and that the first one issatisfied for a given


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We say that a solution v0(t) of (39) is nonuniformly hyperbolic if the linearequation in (7) with

A(t) =F

v(t, v0(t))

admits a nonuniform exponential dichotomy. We continue to assume thatall solutions of equation (7) are global. The following statement is a simple

consequence of Theorem 4.1.Theorem 4.2. Let v0(t) be a nonuniformly hyperbolic solution of equa-tion (39). If there exist constantsc >0 andq >0 such thatFv (t, y+ v0(t)) A(t)

cyqfor every t 0 and y X, and (37) holds, then there exist > 0 and aunique function X such that the set

W ={(s ,,(s, )) + (0, v0(s)) : (s, ) X} (40)

has the following properties:

1. if

(s, vs) {(s ,,(s, )) + (0, v0(s)) : (s, ) X},then(t, v(t)) W for every t s, wherev(t) =v(t, vs) is the uniquesolution of (39) withv(s) =vs;

2. there existsK >0 such that for everys 0, (s, vs), (s, vs) W, andt s we have

v(t, vs) v(t, vs) K ea(ts)+svs vs.

We call to the set W in (40) a Lipschitz stable manifold ofv0(t).

4.2. Smooth stable manifolds. We now discuss the existence of smoothstable invariant manifolds. Theorem 4.1 already shows that the functionin (35) is unique in a class of Lipschitz functions. Thus, to show thatthe set V in (35) is a smooth manifold it is sufficient to show that issmooth. However, it is more convenient to deal with the problem from thebeginning, finding independently a unique function, now in an appropriateclass of smooth functions, that a posteriori must coincide with the function already obtained in Theorem 4.1. The main novelty in our approach, besidesconsidering the nonuniform setting, is the method of proof, which uses asingle fixed point problem to obtain the maximal degree of regularity forthe function. We refer to Section 4.3 for details on the method of proof.

We assume in this section that A : R+0 B(X) and f: R+0 XX are

of class Ck. We continue to denote by Bs() E(s) be the open ball ofradius >0 centered at zero. Given q > k we set

=

(q+ k+ 2)

q k > and =

(2q+ 2)

q k > .We notice that when k = 0 the numbers and coincide respectively with and in (33) and (36). We continue to consider the set X in (34).

We assume that there exist constants c, >0, q > k, and (0, 1] suchthat for every t 0 and u, v Xwe have

f(t, 0) = 0, f

v(t, 0) = 0, f|Y = 0, (41)


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where Y ={(s, v) R+0 X :v

es},jfvj(t, u) cuq+1j for j = 1, . . . , k,

and

kf

vk

(t, u) kf

vk

(t, v) cu v(u + v)qk.

The last condition in (41) may sometimes be obtained with an appropriatecut-off of the function f(see also the discussion after Theorem 4.3).

We denote by Z the space of continuous functions : X Xof classCk in such that for each s 0 and , Bs(e

s) we have

(s, 0) = 0,

(s, 0) = 0, (s, Bs(e

s)) F(s),jj(s, ) 1 for j = 1, . . . , k,

and

k

k(s, )

k

k(s,)

.

For each function Z we consider its graph V given by (35).We now formulate a stable manifold theorem. Set ps, = (s ,,(s, )).

Theorem 4.3 ([9, Theorem 1]). If equation (7)admits a nonuniform expo-nential dichotomy with

a + 0 and a unique function Z such that

(ps,) V for every(s, ) X and0.

Furthermore:

1. V is a manifold of classCk that contains the lineR+

0

{0}and satisfiesT(s,0)V = R E(s) for everys 0;

2. there existsK > 0 such that for everys 0, , Bs(es), and

0 we havejj (ps,) jj

(ps,)

K ea+(j+1)s forj= 0, . . . , k 1, andkk (ps,)

kk

(ps,)

K ea+(k+1)s . (43)We observe that both inequalities in (42) hold when is sufficiently small.

Note that we also establish the exponential decay on the stable manifold ofthe derivatives up to order k 1 of the flow with respect to the initial con-dition (see (43)). We are not aware of any similar result in the literature,even in the case of uniform exponential dichotomies. The exponential de-cay of the derivatives can be understood in the following manner. Whenwe consider higher-order jets (other than the first, which corresponds to thelinear variational equation), the higher-order linearizations maintain the lin-ear partA(t) of the linear variational equation. Thus, the higher-order jets


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possess essentially the same nonuniform exponential dichotomies, althoughin higher-dimensional spaces. This explains the exponential behavior of thederivatives along the stable manifolds.

We now discuss an alternative version of the stable manifold theorem,partly motivated by the desire of eliminating the third condition in (41). Infact this condition requires an exponential decay on time for the function f

and its derivatives. We consider a function f: R+0 X Xof class C

k

such that:

1. f(t, 0) = 0 and fv (t, 0) = 0 for every t 0;2. for some constant >0, and every t 0 and u, v Xwe havejfvj(t, u)

e(k+2)t forj = 0, . . . , k 1and kfvk(t, u)

kf

vk(t, v)

e(k+2)tu v.Let also X be the space of continuous functions

: {(s, ) R+0 X : E(s)} X

of class Ck in such that for each s 0 and , E(s) we have

(s, 0) = 0,

(s, 0) = 0 and (s, E(s)) F(s),

and jj(s, ) j

j(s,)

forj = 0, . . . , k 1.For each X we consider its graph

W ={(s ,,(s, )) : (s, ) R+0 E(s)} R+0 X.

We now formulate a second stable manifold theorem. We continue towrite ps, = (s ,,(s, )).

Theorem 4.4. If equation (7)admits a nonuniform exponential dichotomyanda + < b, then there exist >0 and a unique function X such that(W) W for every0. Furthermore:

1. W is aCk manifold withT(s,0)W= R E(s) for everys 0;

2. there existsK >0 such that for everys 0, , E(s), 0, andj= 0, . . . , k we have

j

j

(ps,) j

j

(ps,) K ea+(j+1)s .

Fork = 1, this statement is Theorem 1 in [17]. The proof for an arbitraryk follows essentially the same steps as in the proof of Theorem 4.3 (seeSection 4.3 for details).

When equation (7) admits a strong nonuniform exponential dichotomy(see Definition 3.2) we also obtain a characterization of the stable manifoldin terms of the exponential growth rate of solutions.


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Theorem 4.5 ([17, Theorem 2]). Assume that equation (7)admits a strongnonuniform exponential dichotomy withb > 0. Givens 0 andv X, if

limsupt+

1

t log t(s, v) b 2,

then(s, v) W.

4.3. Method of proof. Our approach to prove the stable manifold theo-rems (Theorems 4.3 and 4.4) starts by using the differential equation and thedesired invariance of the manifold to conclude that it must be the graph ofa function satisfying a certain fixed point problem (as in many other stablemanifold theorems). However, the extra small exponentials of a nonuniformexponential dichotomy led us to consider two fixed-point problemsone toobtain an a priori bound for the stable component of the solutions alongeach graph, and the other to obtain the graph that gives the stable man-ifold. To obtain the estimates in the fixed point problems, we need sharpbounds for several derivatives, such as for example for the derivatives of thevector field along each graph, i.e., of the vector field

(t, )f(t ,,(t, )).For this we use a multivariate version of the Faa di Bruno formula in [25]for the derivatives of a composition. Consider open sets Y, Z, and W ofBanach spaces. Let g : Y Z be defined in a neighborhood of x Ywith derivatives up to order n at x. Let also f: Z Wbe defined in aneighborhood ofy = g(x) Zwith derivatives up to order n at y. Then-thderivative of the composition h= f g at x is given by

dnxh=n

k=1

dkyf

0r1,...,rknr1++rk=n

cr1rkdr1x g d

rkx g, (44)

for some nonnegative integers cr1rk . The formula allows us to estimate the

norm of the derivatives of a composition in terms of the original functions.Namely, collecting derivatives of equal order, one can show that for eachn N there exists c= c(n)> 0 such that (see [25])

dnxh cn

k=1

dkyf

p(n,k)

nj=1

djxgkj ,

where

p(n, k) =

(k1, . . . , kn) Nn0 :

nj=1

kj =k and

nj=1

jkj =n

(here N0 is the set of nonnegative integers). A multivariate extension ofthe Faa di Bruno formula in (44) was established in [25]. Although severalspecial cases were treated before, the general formula for the derivatives ofa composition was first obtained by Faa di Bruno in [30]. We recommend[46] for the history of the problem and for many related references.

In the proof of the stable manifold theorems we also use a result of Elbialyin [29] (see Theorem 4.6), that goes back to a lemma of Henry in [38], andwhich allows us to establish simultaneouslythe existence and regularity of


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the stable manifolds using a single fixed point problem, instead of one foreach of the successive derivatives. We now briefly describe Henrys lemma.Let X and Y be Banach spaces, and let U X be an open set. Givenconstants (0, 1], k N {0}, and c >0 we define the set

Ck,c (U, Y) = u Ck,(U, Y) :uk, c

,

where Ck,(U, Y ) is the space of Ck functions u : U Y with -Holdercontinuous k-th derivative, and with the norm

uk, = max

u, du, . . . , dku, H(d

ku)

,

where denotes the supremum norm and

H(u) = sup

u(x) u(y)

x y :x, y Xand x=y

.

The following result shows that Ck,c (U, Y) is closed in the space of contin-

uous functions C(U, Y) with the supremum norm.

Theorem 4.6(see [38, 29]). LetXandYbe Banach spaces, and letUX

be an open set. If un Ck,c (U, Y) for each n N and u : U Y is a

function such thatun u 0 asn , thenu Ck,c (U, Y) and for

eachx U we havedkun(x) dku(x) asn .

Whenk = 1 the statement in Theorem 4.6 was first established by Henryin [38, Lemma 6.1.6]. A related result was obtained by Lanford in [44,Lemma 2.5]. In particular, Theorem 4.6 says that the closed unit ball inCk,

is closed with respect to theC0-topology. This property allows us to considercontraction maps solely using the supremum norm, instead of any norm that

also involves the derivatives. For example, it follows in a straightforwardmanner from Theorem 4.6 that the space Z (see Section 4.2) with thenorm

= sup

(s, )

:s 0 and Bs(e

s) \ {0}

(45)

is a complete metric space. The function Zin Theorem 4.3 is preciselyobtained as the unique fixed point of a map in Zthat is a contraction withthe norm in (45).

In [5, 4] we presented an alternative proof of the stable manifold theoremwith slightly weaker assumptions, although only in the finite-dimensionalcase. This is due to the fact that we use in a decisive manner the compactnessof the closed unit ball in Rn. The proof is based on the construction ofan invariant family of cones along each orbit, as in the uniform hyperbolictheory, although now using a family of Lyapunov norms. The family of conesallows us to obtain an invariant distribution that coincides with the tangentspace of the stable manifold. This procedure also allows us to discuss thecontinuity of the distribution, and thus the continuity of the tangent spaces,which corresponds to the C1 regularity of the stable manifold.


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5. Construction of conjugacies

We address in this section the construction of topological conjugaciesbetween the flows defined by the equations

v =A(t)v and v =A(t)v+ f(t, v), (46)

in the general case when the linear equation admits a nonuniform exponen-tial dichotomy. We also show that it is always possible to construct Holdercontinuous conjugacies. We refer to [15] for the construction of topologi-cal (and Holder continuous) conjugacies between nonautonomous dynamicswith distinct linear parts.

5.1. Conjugacies for flows. Let X be a Banach space. We continue todenote byB(X) the set of bounded linear operators in X, and we consider acontinuous function A : R B(X) such that each solution of (7) is definedfor everyt R. We also consider a continuous functionf: R XXandwe assume that there exists > 0 such that for every t R and x, y Xwe have

f(t, x) f(t, y) e6|t|

min{1, x y}(with the constant as in (22)). The equations in (46) define evolutionoperators that we denote respectively by T(t, s) and R(t, s), for t, s R.

Theorem 5.1 ([15, Theorem 7]). If equation (7) admits a strong nonuni-form exponential dichotomy inRwithb >0, andis sufficiently small, thenthere exist homeomorphismsht : XX fort R such that

T(t, s) hs = ht R(t, s), t, s R.

It is also shown in [15] that the maps ht are unique among those thatare close to the identity for each t. More precisely, the maps ht Id areuniformly bounded in t with respect to a family of Lyapunov norms. Werefer to Section 5.2 for a detailed discussion in the case of discrete time.

The original references for the GrobmanHartman theorem (in the case ofuniformly hyperbolic dynamics) are Grobman [32, 33] and Hartman [36, 37].Using the ideas in Mosers proof in [52] of the structural stability of Anosovdiffeomorphisms, the GrobmanHartman theorem was extended to Banachspaces independently by Palis [55] and Pugh [64]. In the case of continu-ous time a version of the GrobmanHartman theorem for nonautonomousdifferential equations v = A(t)v was obtained by Palmer in [56] (with theexception of the Holder property of the conjugacies), although assuming theexistence of a uniform exponential dichotomy.

We can also show that the topological conjugacieshtand their inverses are

Holder continuous. We assume that equation (7) admits a strong nonuniformexponential dichotomy and we set

0= min{a/a, b/b}.

It follows from (22) that 0 (0, 1].

Theorem 5.2 ([15, Theorem 7]). If equation (7) admits a strong nonuni-form exponential dichotomy inR withb >0, then for each (0, 0) and


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eachsufficiently small (depending on), there exist homeomorphisms htas in Theorem 5.1, and a constantK >0 (depending on and) such that

ht(x) ht(y) K e2(2+3)|t|x y

andh1t (x) h

1t (y) K e

2(2+3)|t|x y

for every t R andx, y X withx y e3|t|.

We note that in the case of uniform exponential dichotomies, the H olderregularity of the conjugacies seems to have been known by the experts forsome time, although the first proof only appeared recently in [11]. TheHolder property was claimed by van Strien in [79, Proposition 4.6], but itwas observed in [67] (see also [34]) that some problems in the proof are notyet overcome. This should be compared with the discussion in [34], wherethe authors indicate that the statement of the Holder regularity is containedin a preprint of Belitski [19] (the preprint apparently circulates since 1994but it remains unpublished).

5.2. Conjugacies for maps. The conjugacies for flows are constructed byfirst considering the time-1 maps. After obtaining conjugacies for thesemaps, we use a general integrating procedure to obtain the desired con-jugacies for the original flows.

We first consider arbitrary maps (not necessarily obtained from a flow).For each m Zconsider:

1. an invertible linear operator Am B(X) with inverse A1m B(X);

2. a continuous map fm : XX.

We now introduce the notion of nonuniform exponential dichotomy in thecase of discrete time. Set

A(m, n) =Am1 An, m > n.

Definition 5.3. We say that the sequence of linear operators (Am)mZadmits a nonuniform exponential dichotomyif there exist projections Pn B(X) for n Z with

PmA(m, n) = A(m, n)Pn for every m, n Z with m n,

and there exist constants

D >0, a < 0 b, and 0 (47)

such that for every m, n Z with m n we have

A(m, n)Pn Dea(mn)+|n|, A(m, n)1Qm De

b(mn)+|m|,

where Qn= Id Pn are the complementary projections.

For a sequence (Am)mZ admitting a nonuniform exponential dichotomy,we consider the linear subspaces

Em= PmX and Fm=QmX

for m Z. Clearly, X = Em Fm for every m Z, and the dimensionsdim Em and dim Fm are independent ofm. We also define the operators

Bm= Am|Em :Em Em+1 and Cm = Am|Fm : FmFm+1


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for m Z. Clearly, Bm and Cm are invertible continuous linear operatorswith continuous inverse. Furthermore, with respect to the decompositionsX=Em Fm each operator Am has the form

Am=

Bm 0

0 Cm

, m Z.

Setting, for each m n,B(m, n) =Bm1 Bn and C(m, n) =Cm1 Cn,

we introduce Lyapunov norms as follows. Choose > 0 such that 0 such that for each m Z andx, y Xwe have

fm(x) e|m|

andfm(x) fm(y) e

|m|x y

(with the constant as in (47)).

The following is a version of the GrobmanHartman theorem in thisnonuniformly hyperbolic setting.

Theorem 5.4 ([8, Theorems 1 and 3 and Corollary 1]). If the sequence(Am)mZ admits a nonuniform exponential dichotomy with b > 0, and is sufficiently small, then there exist unique sequences (um)mZ X and(vm)mZ X such that for everym Z we have

Amum=um+1 (Am+ fm)and

vm+1 Am = (Am+ fm)

vm,

where

um= Id +um and

vm = Id +vm. Furthermore,umvm=vmum = Id, m Z.The Holder regularity of the conjugacies is also established in [8].We now explain how we can obtain the conjugacies for the flows from

the conjugacies for the time-1 maps. Fixr [1, 1]. For each m Z weconsider the invertible linear operators

Am= T(m + r, m+ r 1),


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and the continuous maps

fm(u) =

m+rm+r1

T(m + r, )f(, v(, u)) d,

where v(t, u) is the solution of the differential equation v =A(t)v+ f(t, v)with v(m+r 1) = u. One can show in a simple manner that when the

conditions in Theorem 5.1 hold we can apply Theorem 5.4 (in particular,the sequence of operators (Am)mZ admits a nonuniform exponential di-chotomy). Therefore, there exist homeomorphisms Gm,r :XXform Zsuch that

T(m+ r, m+ r 1) Gm,r =Gm+1,r R(m + r, m+ r 1).

Furthermore, the uniqueness statement in Theorem 5.4 can be used to showthat

Gm,r =Gm,r whenever m + r= m + r.

In other words, we can define homeomorphisms

Jt= Gm,r with m= [t] and r = t [t],

where [t] denotes the integer part oft. Finally, for each tR

we define themap ht : XX by

ht(x) =

10

[T(t, r+ t) J+t R(+ t, t)](x) d.

One can show that these maps are the required conjugacies, with the prop-erties in Theorems 5.1 and 5.2.

6. Center manifolds and reversibility

This section is dedicated to the construction of invariant center manifolds.We also discuss their reversibility properties when the ambient dynamics isreversible.

6.1. Existence of center manifolds. LetA : R B(X) be a continuousfunction. We continue to assume that all solutions of equation (7) are global.We first introduce the notion of nonuniform exponential trichotomy.

Definition 6.1. We say that equation (7) admits a nonuniform exponentialtrichotomy if there exist functions P, Q1, Q2 : R B(X) such that P(t),Q1(t), and Q2(t) are projections with

P(t) + Q1(t) + Q2(t) = Id,

P(t)T(t, s) =T(t, s)P(s), Qi(t)T(t, s) =T(t, s)Qi(s), i= 1, 2

for every t, s R, and there exist constants

D, 0, 0 a < b, and 0 c < d (48)

such that:

1. for every s, t Rwith t s we have

T(t, s)P(s) Dea(ts)+|s|, T(t, s)1Q2(t) Deb(ts)+|t|;

2. for every s, t Rwith t s we have

T(t, s)P(s) Dec(st)+|s|, T(t, s)1Q1(t) Ded(st)+|t|.


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In a certain sense, the existence of a nonuniform exponential trichotomyis the weakest hypothesis under which one is able to establish the existenceof invariant center manifolds.

We now present the hypotheses for the nonlinear part of the vector field.We assume that there is an integer k N such that A : R B(X) andf: R XXare of class Ck. We also assume that:

1. f(t, 0) = 0 and fv (t, 0) = 0 for every t R;2. there exists >0 such that for every t Rand u, v Xwe havejfvj(t, u)

e(k+2)|t| forj= 1, . . . , k,and kfvk(t, u)

kf

vk(t, v)

e(k+2)|t|u v.In the presence of a nonuniform exponential trichotomy we consider the

subspaces

E(t) =P(t)X, F1(t) =Q1(t)X, and F2(t) =Q2(t)X.

Let Xbe the space of continuous functions

= (1, 2) :{(s, ) R X: E(s)} X

of class Ck in such that for every s R and , E(s) we have

(s, 0) = 0,

(s, 0) = 0, (s, E(s)) F1(s) F2(s),

jj(s, ) 1 for j = 1, . . . , k ,

and

k

k(s, )

k

k(s,) .

Given a function X we consider its graph

V ={(s ,,(s, )) : (s, ) R E(s)} R X. (49)

We now present the center manifold theorem. We continue to use thenotation ps, = (s ,,(s, )), and we set = 4cD. We also denote by the flow generated by the autonomous equation in (31).

Theorem 6.2 ([14, Theorem 3]). If equation (7) admits a nonuniform ex-ponential trichotomy with

(k+ 1)(a+ ) b


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2. there existsK >0 such that for eachs R, , E(s), R, andj= 0, . . ., k we have

jj

(ps,) j

j (ps,)

K e(j+1)[(a+)+|s|] , 0,andjj

(ps,) j

j (ps,)

K e(j+1)[(c+)||+|s|] , 0.The proof of Theorem 6.2 follows a similar strategy to the one described

in Section 4.3 in the case of stable manifolds, that is based in a lemma ofHenry and in the Faa di Bruno formula to estimate the derivatives of acomposition.

The study of center manifolds can be traced back to the works of Pliss [61]and Kelley [42]. A very detailed exposition in the case of autonomous equa-tions is given in [76], adapting results in [78]. See also [51, 77] for the case ofequations in infinite-dimensional spaces. We refer to [20, 23, 24, 76] for more

details and further references. We emphasize that all these works consideronly the case of uniformly hyperbolic dynamics.

6.2. Reversibility in center manifolds. We now show that the (time) re-versibility of a given flow descends to the center manifold obtained in Theo-rem 6.2. We recall that time-reversal symmetries are among the fundamentalsymmetries in many physical systems. In particular, many Hamiltoniansystems are reversible (see [43] for several examples).

We first introduce the notion of reversible (nonautonomous) differentialequation. LetXbe a Banach space, and let L : R XXbe a continuousfunction such that

v =L(t, v) (50)

has unique and global solutions. We also consider a (Frechet) differentiablemap S: R XX.

Definition 6.3. We say that equation (50) is reversiblewith respect to Sif

L(t, S(t, v)) +S

v(t, v)L(t, v) =

S

t(t, v)

for every t Rand v X.

We now describe a characterization of reversibility. In fact, this char-acterization can be seen as the main justification for the above notion ofreversible equation. For each s R and vs X, we denote by (t, s)(vs)

the unique solution of (50) with v(s) =vs. We recall that by hypothesis theoperator (t, s) is defined for every t, s R.

Theorem 6.4 ([13, Proposition 1]). Equation(50)is reversible with respectto the map Sif and only if

(, t)(S(t, v)) =S(, (, t)(v))

for every t, R andv X.


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We now formulate our result about the reversibility in center manifolds,showing that the reversibility of a given equation, with respect to a mapS with St = S(t, ) linear for each t R, always descends to the centermanifold. We assume that for some constants C >0 and 0 we have

C1e|t| S1t 1 St C e

|t|, t R. (51)

We also writeVs = {v X : (s, v) V}= {(, (s, )) : E(s)},

where is the function given by Theorem 6.2.

Theorem 6.5 ([13, Theorem 2]). Under the assumptions of Theorem 6.2,if the equation

v =A(t)v+ f(t, v)

is reversible with respect to a map S with S20 = Id and St linear for eacht R, and the constants in (48) and (51) satisfy

max{c, a} + 2( + )< min{b, d},

thenSs(Vs) = Vs for everys R.

7. Lyapunov regularity

We present briefly in this section the abstract theory of Lyapunov expo-nents, and its associated regularity theory developed by Lyapunov in hisdoctoral thesis [47]. We also explain how one can use this theory to esti-mate the constants in the notion of nonuniform exponential dichotomy. Atpresent this theory is only fully developed for finite-dimensional spaces. Ageneralization to infinite dimension is given in [6] in the particular case ofHilbert spaces.

7.1. Lyapunov exponents and regularity. Consider matricesA(t) vary-ing continuously with t 0. We assume that

limsupt+

1

t log+A(t)= 0,

where log+ x= max{0, log x}(we use the convention that log 0 =).

Definition 7.1. We define the Lyapunov exponent : Rn R {}associated to equation (7) by

(v0) = lim supt+

1

t logv(t),

where v(t) is the solution of (7) with v(0) =v0

.

One can easily show that the following properties hold (see [1] for details):

1. (v) =(v) for each v Rn and R \ {0};2. (v+ w) max{(v), (w)} for each v, w Rn;3. (v+ w) = max{(v), (w)} for each v, w Rn with (v)=(w);4. given v1, . . . , vm R

n \ {0}, if the numbers (v1), . . ., (vm) aredistinct, then the vectors v1, . . . , vm are linearly independent.


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By the last property, the function takes at most p n distinct values onRn \ {0}, say

1 < < p.

Moreover, by the first two properties, for i= 1, . . . , pthe set

Ei = {v0 Rn :(v0) i} (52)

is a linear subspace. Note that (v0)> i for every v0 Rn \ Ei.To introduce the notion of Lyapunov regularity we consider the initial

value problem of the adjoint equation

w =A(t)w, w(0) =w0, (53)

with w0 Rn, where A(t) denotes the transpose ofA(t). We also consider

the corresponding Lyapunov exponent : Rn R {}defined by

(w0) = lim supt+

1

t logw(t),

where w(t) is the solution of (53).We denote by, the canonical inner product in Rn. We recall that two

basesv1, . . . , vn andw1, . . . , wn ofRn are said to bedual ifvi, wj= ij forevery i and j , where ij is the Kronecker symbol.

Definition 7.2. Theregularity coefficientof the pair of Lyapunov exponents and is defined by

(, ) = min max{(vi) + (wi) : 1 i n},

where the minimum is taken over all dual bases v1, . . . , vn and w1, . . . , wnof Rn. We say that equation (7) is Lyapunov regular or simply regular if(, ) = 0.

We note that the notion of regularity can also be expressed solely usingequation (7), thus without the need for the adjoint equation in (53). Namely,(7) is regular if and only if

limt+

1

t

t0

tr A() d=

pi=1

(dim Ei dim Ei1)i,

setting E0 = {0} (we refer to [1] for details). However, it is sometimesconvenient to use the above description of regularity, in terms of(, ), forexample in the proof of Theorem 7.3.

7.2. Existence of nonuniform exponential dichotomies. We assumein this section that there is at least one negative Lyapunov exponent forequation (7), i.e., that for some 1 k p we have

1< < k


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for everyt 0. We also consider the Lyapunov exponents associated to theblocks B(t) and C(t) in (54), i.e., associated to the pair of equations

x =B(t)x and x =B(t)x,

and to the pair of equations

y =C(t)y and y =C(t)y.

The corresponding regularity coefficients are

1= (|E, |E) and 2= (|F, |F). (55)

The following result shows that any linear differential equation as aboveadmits a nonuniform exponential dichotomy, and in fact a strong nonuni-form exponential dichotomy. Furthermore, the constants in the notion ofdichotomy can be related to the Lyapunov exponents and the regularitycoefficients.

Theorem 7.3 ([12, Theorem 4]). Assume that the matricesA(t) have theform in (54) for everyt 0, and that equation (7) has at least one negativeLyapunov exponent. Then, for each > 0, equation (7) admits a strong

nonuniform exponential dichotomy witha= 1+ , a= k+ , b= k+1+ , b= p+ ,

and= max{1, 2} + 2.

By Theorem 7.3, the deviation of a nonuniform exponential dichotomywith respect to a uniform dichotomy is measured by the regularity coeffi-cients in (55). It is therefore of interest to obtain sharp bounds for(, ), ifpossible expressed only in terms of the matrices A(t), thus without knowingexplicitly the solutions of equation (7).

We first present a lower bound for the regularity coefficient.

Theorem 7.4 ([12, Theorem 5]). We have

(, ) 1nlimsup

t+1t t

0tr A() d lim inf

t+1t t

0tr A() d .

We now describe a geometric consequence of Theorem 7.4. Let v1, . . .,vn be a basis ofR

n. We denote by n(t) the n-volume of the parallelepipeddefined by the vectorsv1(t),. . .,vn(t), wherevi(t) is the solution of (7) withv0 = vi, for i= 1, . . ., n. By the Liouville formula we have

n(t)/n(0) = exp

t0

tr A() d,

and it follows from Theorem 7.4 that

lim supt+

1

t log n(t) lim inf

t+

1

t log n(t) n (, ).

In particular, when equation (7) is regular we have

limt+

1

t log n(t) = lim

t+

1

t

t0

tr A() d

(and the limits are independent of the basis v1, . . ., vn).We now present an upper bound for the regularity coefficient. We only

consider triangular matrices, for which we can describe the results more


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explicitly (we refer to [12] for details on the reduction to the triangularcase). For k = 1, . . . , n, we consider the numbers

k = lim inft+

1

t

t0

ak() d and k = lim supt+

1

t

t0

ak() d,

where a1(t), . . ., an(t) are the entries in the diagonal ofA(t).

Theorem 7.5 ([12, Theorem 6]). If the matricesA(t) are upper triangularfor every t 0, then

(, )n

k=1

(k k).

It follows from Theorems 7.4 and 7.5 that in the 1-dimensional case, i.e.,for a scalar equation v =a(t)v the regularity coefficient is given by

(, ) = lim supt+

1

t

t0

a() d lim inft+

1

t

t0

a() d.

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Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lisboa,

Portugal

E-mail address: [email protected]: http://www.math.ist.utl.pt/~barreira/

Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lisboa,

Portugal

E-mail address: [email protected]

stability in nonautonomous dynamics - barreira

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