ON THE SUSPENSION ISOMORPHISM FOR INDEX BRAIDS IN A SINGULAR PERTURBATION PROBLEM

Author(s)

Carbinatto, Maria C. ^[1] ; Rybakowski, Krzysztof P. ^[2]

Affiliation

^[1] ICMC USP, Dept Matemat, BR-13560970 Sao Carlos, SP - Brazil

^[2] Univ Rostock, Inst Math, D-18055 Rostock - Germany

• Document type: Journal article
• Source: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS; v. 32, n. 2, p. 199-225, DEC 2008.
• Web of Science Citations: 1

Abstract

We consider the singularly perturbed system of ordinary differential equations (E(epsilon)) epsilon(y)over dot = f(y, x, epsilon), (x)over dot = h(y, x, epsilon) on Y x M, where Y is a finite dimensional normed space and M is a smooth manifold. We assume that there is a reduced manifold of (E(epsilon)) given by the graph of a function phi: M -> Y and satisfying an appropriate hyperbolicity assumption with unstable dimension k is an element of N(0). We prove that every Morse decomposition (M(p))(p is an element of P) of a compact isolated invariant set S(0) of the reduced equation (x) over dot = h(phi(x), x, 0) gives rises, for epsilon > 0 small, to a Morse decomposition (M(p,) (epsilon))(p is an element of P) of an isolated invariant set S(epsilon) of (E(epsilon)) such that (S(epsilon), (M(p,) (epsilon))(p is an element of P)) is close to ([0] x S(0), ([0] x M(p))(p is an element of P)) and the (co)homology index braid of (S(epsilon), (M(p,) (epsilon))(p is an element of P)) is isomorphic to the (co)homology index braid of (S(0), (M(p))(p is an element of P)) shifted by k to the left. (AU)