Smoothness of Invariant Manifolds for Nonautonomous Equations

Barreira, Luís; Valls, Clàudìa

doi:10.1007/s00220-005-1380-z

Cited by 28 publications

(55 citation statements)

References 11 publications

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“…It is shown in [4] that any equation v = A(t)v with A(t) as above admits a nonuniform exponential dichotomy. Thus, in specific applications, such as in Theorem 1, we never need to assume the existence of a nonuniform exponential dichotomy (since this is always the case), but instead we look for "smallness" conditions on a and b which ensure the desired results.…”

Section: Theorem 1 Assume That Eqmentioning

confidence: 98%

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Stable manifolds for nonautonomous equations without exponential dichotomy

Barreira¹,

Valls²

2006

Journal of Differential Equations

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We consider nonautonomous ordinary differential equations v = A(t)v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation (t, v), which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v = A(t)v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.

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Section: Theorem 1 Assume That Eqmentioning

confidence: 98%

“…If the conditions q + a < 0 and + b < (4) are satisfied, then there exists a Lipschitz manifold W ⊂ R × X which is the graph of a Lipschitz function : U → F , where U ⊂ R + 0 × E is an open neighborhood of the line R + 0 × {0}, and the following properties hold: (1) (t, 0) ∈ W for every t 0;…”

Section: Theorem 1 Assume That Eqmentioning

confidence: 99%

Stable manifolds for nonautonomous equations without exponential dichotomy

Barreira¹,

Valls²

2006

Journal of Differential Equations

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“…This is certainly the case when (2) admits an exponential dichotomy: by an appropriate version of the Grobman-Hartman theorem, and under certain "smallness" assumptions on the perturbation, locally the two dynamics are topologically conjugate (we refer to [3] for detailed references; we note that [3] also considers the more general case of nonuniform exponential dichotomies). When Eq.…”

Section: Motivationmentioning

confidence: 99%

“…(1). In particular, we do not require the linear equation (2) to possess a uniform exponential behavior (either in the central, stable, or unstable directions). We still use some amount of partial hyperbolicity to establish the existence of the center manifolds, but this hyperbolicity can be spoiled exponentially along each solution as the initial time changes.…”

Section: Nonuniform Exponential Behaviormentioning

confidence: 99%

“…We note that a principal motivation for weakening the assumption of uniform exponential behavior for Eq. (2) is that almost all trajectories of a dynamical system preserving an invariant measure on a smooth compact manifold have a linear variational equation which admits a weak nonuniform exponential trichotomy (see [2] for more details).…”

Section: Nonuniform Exponential Behaviormentioning

confidence: 99%

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Smooth center manifolds for nonuniformly partially hyperbolic trajectories

Barreira

Valls

2007

Journal of Differential Equations

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We establish the existence of unique smooth center manifolds for ordinary differential equations v = A(t)v + f (t, v) in Banach spaces, assuming that v = A(t)v admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the C k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting.

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Stability of Nonautonomous Differential Equations

Barreira¹,

Valls²

2008

Lecture Notes in Mathematics

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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Typesetting by the authors and SPi using a Springer L A T E X macro package Cover design: design & production GmbH, HeidelbergPrinted on acid-free paper SPIN: 12114993 41/SPi 5 4 3 2 1 0To our parents PrefaceThe main theme of this book is the stability of nonautonomous differential equations, with emphasis on the study of the existence and smoothness of invariant manifolds, and the Lyapunov stability of solutions. We always consider a nonuniform exponential behavior of the linear variational equations, given by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy. Thus, the results hold for a much larger class of systems than in the "classical" theory of exponential dichotomies. The departure point of the book is our joint work on the construction of invariant manifolds for nonuniformly hyperbolic trajectories of nonautonomous differential equations in Banach spaces. We then consider several related developments, concerning the existence and regularity of topological conjugacies, the construction of center manifolds, the study of reversible and equivariant equations, and so on. The presentation is self-contained and intends to convey the full extent of our approach as well as its unified character. The book contributes towards a rigorous mathematical foundation for the theory in the infinite-dimensional setting, also with the hope that it may lead to further developments in the field. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.The first part of the book serves as an introduction to the other parts. After giving in Chapter 1 a detailed introduction to the main ideas and motivations behind the theory developed in the book, together with an overview of its contents, we introduce in Chapter 2 the concept of nonuniform exponential dichotomy, which is central in our approach, and we discuss some of its basic properties. Chapter 3 considers the problem of the robustness of nonuniform exponential dichotomies.In the second part of the book we discuss several consequences of local nature for a nonlinear system when the associated linear variational equation admits a nonuniform exponential dichotomy. In particular, we establish in Chapter 4 the existence of Lipschitz stable manifolds for nonautonomous equations in a Banach space. In Chapters 5 and 6 we establish the smooth-VIII Preface ness of the stable manifolds. We first consider the finite-dimensional case in Chapter 5, with the method of invariant families of cones. This approach uses in a decisive manner the compactness of the closed unit ball in the ambient space, and this is why we consider only finite-dimensional spaces in this chapter. Moreover, ...

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Smoothness of Invariant Manifolds for Nonautonomous Equations

Cited by 28 publications

References 11 publications

Stable manifolds for nonautonomous equations without exponential dichotomy

Stable manifolds for nonautonomous equations without exponential dichotomy

Smooth center manifolds for nonuniformly partially hyperbolic trajectories

Stability of Nonautonomous Differential Equations

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