Stability in nonautonomous dynamics: a survey of recent results (Q2390128)

In this comprehensive survey the authors describe recent results on nonautonomous differential equations, mostly concentrating on their previous work. More exactly, the main theme of the work is the stability of nonautonomous differential equations under sufficiently small nonlinear perturbations, with emphasis on the study of the Lyapunov stability of solutions and of the existence and smoothness of invariant manifolds. Section 2 is devoted to the study of the persistence of asymptotic stability of a nonuniform exponential contraction under a sufficiently small nonlinear perturbation. The authors first present some examples on systems with nonuniform exponential contractions. Next they refer to some results from their paper entitled ``Nonuniform exponential dichotomies and Lyapunov regularity'' [J. Dyn. Differ. Equations 19, No. 1, 215--241 (2007; Zbl 1123.34040)] according to which the existence of nonuniform exponential contractions for systems is guaranteed, provided that all Lyapunov exponents are negative. Section 3 deals with the general case of nonuniform exponential dichotomies concerning the linear case theory. In particular, the authors discuss the robustness of nonuniform exponential dichotomies, namely, whether a linear equation with a nonuniform exponential dichotomy has the property that sufficiently small linear perturbations admit such dichotomies. The main existence results of this section is given in a theorem borrowed from the paper of the authors entitled ``Robustness of nonuniform exponential dichotomies in Banach spaces'' [J. Differ. Equations 244, No. 10, 2407--2447 (2008; Zbl 1158.34041)]. In Section 4 the authors investigate the construction of stable invariant manifolds for sufficiently small perturbations of linear differential equations admitting a nonuniform exponential dichotomy. They examine what happen for the stability of nonuniform exponential contractions, when not all directions are contracting. The existence of stable invariant manifolds corresponds to the existence of asymptotic stability along a certain invariant manifold, which is tangent to the stable space. Section 5 is devoted to the construction of topological conjugacies between the flows defined by the equations \(v' = A(t)v\) and \(v' = A(t)v + f(t, v),\) in the general case when the linear equation admits a nonuniform exponential dichotomy. The authors show that it is always possible to construct Hölder continuous conjugacies. To show this they refer to some results from their (unpublished yet) paper, ``Conjugacies for linear and nonlinear perturbtaions of nonuniform behavior'' [J. Funct. Anal., to appear], as well as from their paper ``Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy'' [J. Funct. Anal. 238, No. 1, 118--148 (2006; Zbl 1099.37020)] for the construction of topological (and Hölder continuous) conjugacies between nonautonomous dynamics with distinct linear parts. In Section 6 the authors discuss the construction of invariant center manifolds, provided that the equation admits a nonuniform exponential trichotomy. They also discuss their reversibility properties when the ambient dynamics is reversible. Finally, in Section 7, conditions are given for equation \(v' = A(t)v\) to admit a strong nonuniform exponential dichotomy. The results are borrowed from the authors' paper used, also, in Section~2.