Nonautonomous dissipative dynamical systems with a hyperbolic subset of the center (one-dimensional case)

Abstract

We study the structure of the Levinson Center [I] for the scalar dissipative equation u' = f(t, u) with recurrent (almost-periodic, m-periodic) right-hand side. Although the problem under study concerns the equation u' = f(t, u), its solution is given within the framework of general nonautonomous dynamical systems. The approach to nonautonomous differential equations from the point of view of dynamical systems was suggested and developed in the mid sixties by V.M. Millionshchikov, B. A. Sheherbakov, R. K. Miller, J. Seifert, and J. P. Sell, and later on by V. V. Zhikov and others. Everywhere below we will use the terminology and notation, generally adopted in the topological theory of dynamical systems [2-4], and also introduced in [i, 5-7]. Let us recall some of them. Let T be the group of real numbers (R) or of integers ( Z ), T+ = {t]t~ T,t>~O}, be analogously defined, V be a compact metric space, (X, h, Y) be a finite-dimensional vector bundle with the fiber R ~, and ['I be a Riemannian metric on (X, h, Y) that is compatible with the metric p on X, (X, T , ~) and (Y, T, o) be dynamical systems on X and Y, respectively, and