Stability of Nonautonomous Differential Equations

ISBN-13: 9783540747741 / Angielski / Miękka / 2007 / 291 str.

Claudia Valls

Opis

The main theme of this book is the stability of nonautonomous di?erential equations, with emphasis on the study of the existence and smoothness of invariant manifolds, and the Lyapunov stability of solutions. We always c- sider a nonuniform exponential behavior of the linear variational equations, given by the existence of a nonuniform exponential contraction or a nonu- form exponential dichotomy. Thus, the results hold for a much larger class of systems than in the "classical" theory of exponential dichotomies. Thedeparturepointofthebookisourjointworkontheconstructionof- variant manifolds for nonuniformly hyperbolic trajectories of nonautonomous di?erential equations in Banach spaces. We then consider several related - velopments, concerningtheexistenceandregularityoftopologicalconjugacies, the construction of center manifolds, the study of reversible and equivariant equations, and so on. The presentation is self-contained and intends to c- vey the full extent of our approach as well as its uni?ed character. The book contributes towards a rigorous mathematical foundation for the theory in the in?nite-dimensional setting, also with the hope that it may lead to further developments in the ?eld. The exposition is directed to researchers as well as graduate students interested in di?erential equations and dynamical systems, particularly in stability theory.

Recenzje

From the reviews: "In this book, the authors give a unified presentation of a substantial body of work which they have carried out and which revolves around the concept of nonuniform exponential dichotomy. ... This is a well-written book which contains many interesting results. The reader will find significant generalizations of the standard invariant manifold theories, of the Hartman-Grobman theorem ... . Anyone interested in these topics will profit from reading this book." (Russell A. Johnson, Mathematical Reviews, Issue 2010 b)

Spis treści

Exponential dichotomies.- Exponential dichotomies and basic properties.- Robustness of nonuniform exponential dichotomies.- Stable manifolds and topological conjugacies.- Lipschitz stable manifolds.- Smooth stable manifolds in Rn.- Smooth stable manifolds in Banach spaces.- A nonautonomous Grobman–Hartman theorem.- Center manifolds, symmetry and reversibility.- Center manifolds in Banach spaces.- Reversibility and equivariance in center manifolds.- Lyapunov regularity and stability theory.- Lyapunov regularity and exponential dichotomies.- Lyapunov regularity in Hilbert spaces.- Stability of nonautonomous equations in Hilbert spaces.

O autorze

Luis Barreira is a Full Professor of Mathematics at Instituto Superior Técnico, Lisbon and a member of the Center for Mathematical Analysis, Geometry, and Dynamical Systems. He obtained his PhD from the Pennsylvania State University in 1996. In 2007 he has been awarded the Gulbenkian Science Prize.  

Claudia Valls is an Invited Assistant Professor at Instituto Superior Técnico, Lisbon and a Postdoctoral Fellow at the Center for Mathematical Analysis, Geometry, and Dynamical Systems, of which she is also a member. She obtained her PhD from the Universitat de Barcelona in 1999. 

Blurb

Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the study of center manifolds, as well as their reversibility and equivariance properties. Most results are obtained in the infinite-dimensional setting of Banach spaces. Furthermore, the linear variational equations are always assumed to possess a nonuniform exponential behavior, given either by the existence of a nonuniform exponential contraction or a nonuniform exponential dichotomy. The presentation is self-contained and has unified character. The volume contributes towards a rigorous mathematical foundation of the theory in the infinite-dimension setting, and 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.