Luis T. Magalhães

Dynamics in infinite dimensions

Preface.- Introduction.- Invariant Sets and Attractors.- Functional Differential Equations on Manifolds.- The Dimension of the Attractor.- Stability and Bifurcation.- Stability of Morse-Smale Maps and Semiflows.- One-to-oneness, Persistence and Hyperbolicity.- Realization of Vector Fields and Normal Forms.- Attractor Sets as C1-Manifolds.- Monotonicity.- The Kupka-Smale Theorem.- Appendix A: Conley Index Theory in Noncompact Spaces.- References.- Index.

Stability of Morse-Smale Maps

We will deal in this section with smooth maps f: B → E, B being a Banach manifold imbedded in a Banach space E. The maps f belong to Cr(B, E), the Banach space of all E-valued Cr-maps defined on B which are bounded together with their derivatives up to the order r ≥ l. Let Cr(B, B) be the subspace of Cr(B, E) of all maps leaving B invariant, that is, f(B) ⊂ B. Denote by A(f) the set

Stability of Morse—Smale Maps and Semiflows

\begin{array}{*{20}{c}} {A\left( f \right) = \left\{ {x \in B:\,there\;exists\,a\,sequence} \right.\,\left( {x = {{x}_{1}},{{x}_{2}}, \ldots } \right) \in B,} \\ {\left. {\mathop{{\sup }}\limits_{j} \left\| {{{x}_{j}}} \right\| < \infty \;and\,f\left( {{{x}_{j}}} \right) = {{x}_{{j - 1}}},j = 2,3, \ldots } \right\}.} \\ \end{array}

Realization of Vector Fields and Normal Forms

It is of some interest to determine when the set A(F) of an RFDE F is a C 1-manifold, since it will then have a particularly simple geometric structure which will facilitate the study of qualitative properties of the flow. Results in this direction can be established through the use of C k -retractions which are defined as C k maps Υ from a Banach manifold into itself such that Υ2=Υ, k ≥ 1.

The Kupka—Smale Theorem

We start this Chapter by dealing with maps and Section 6.1 corresponds, precisely, to Section 10 of [87]. Later, in Section 6.2, we will present the case of semiflows. Let us consider smooth maps f: B → E, B being a Banach manifold embedded in a Banach space E. The maps ƒ belong to the C r (B, E) Banach space of all E-valued C r-maps defined on B which are bounded together with their derivatives up to the order r ≥ 1. Let C r (B,B) be the subspace of C r (B,E) of all maps leaving B invariant, that is, f(B) I B. Denote by A(f) the set

Invariant Sets and Attractors

The purpose of this section is to present results on the “size” of the attractor. This will be given in terms of limit capacity and Hausdorff dimension. The principal results are applicable not only to RFDE but also to some of the abstract dynamical systems considered in Section 2.

Functional Differential Equations on Manifolds

The persistence of a (compact) normally hyperbolic manifold and the existence of its stable and unstable manifolds, corresponding to the action of C r maps or semiflows, r ≥ 1, are well known facts (see, for instance, [102] and [176]). The more general case of (compact) hyperbolic sets that are invariant under (not necessarily injective) maps was also carefully studied (see Ruelle [176], Shub [189] and Palis and Takens [163]). The persistence and smoothness of (compact) hyperbolic invariant manifolds for RFDE were considered in detail in Magalhaes [132] based on skew-product semiflows defined for RFDE, locally around hyperbolic invariant manifolds, and their spectral properties, associated with exponential dichotomies, following the lines of work developed by Sacker and Sell for flows([186], [187]).