Barreira L., Valls C. Stability of Nonautonomous Differential Equations

Berlin: Springer, 2008. — 283 p.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.Exponential dichotomies and basic properties
Robustness of nonuniform exponential dichotomiesLipschitz stable manifolds
Smooth stable manifolds in Rn
Smooth stable manifolds in Banach spaces
A nonautonomous Grobman–Hartman theoremCenter manifolds in Banach spaces
Reversibility and equivariance in center manifoldsLyapunov regularity and exponential dichotomies
Lyapunov regularity in Hilbert spaces
Stability of nonautonomous equations in Hilbert spaces