Stability of Nonautonomous Differential Equations

Abstract

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, ...

A Perron-type theorem for nonautonomous delay equations

“…We refer the reader to the books [1,2] for details and references (also for the infinite-dimensional setting).…”

Polynomial stability of evolution operators in Banach spaces

“…Characterizations for exponential stability properties of strongly measurable evolution operator with uniform growth are given in [1,6,7] for u.e.s, respectively in [4,8] and [10] for e.s, respectively in [2,3,8] and [9] for B.V.e.s.…”

“…The concepts of uniform exponential stability and nonuniform exponential stability are well-known and the concept of exponentially stable in the Barreira-Valls sense has been considered in the works of L. Barreira and C. Valls, as for example [2] and [3].…”

Smooth stable invariant manifolds on time scales

“…As one of the most important and useful properties, the classical theory of invariant manifolds provides a geometric structure to describe and understand the qualitative behavior of nonlinear dynamical systems and has been widely recognized both in mathematics and in applications [18]. In particular, with the help of nonuniform exponential dichotomies, Zhang, Fan and Chang [6] investigated the existence of Lipschitz stable invariant manifolds for nonuniformly hyperbolic systems on measure chains.…”