On the suspension isomorphism for index braids in a singular perturbation problem

Abstract

We consider the singularly perturbed system of ordinary differential equations \begin{equation} \begin{split} \varepsilon\dot y&=f(y,x,\varepsilon), \\ \dot x&=h(y,x,\varepsilon) \end{split} \tag{$(E_\varepsilon)$} \end{equation} on $Y\times \mathcal{M}$, where $Y$ is a finite dimensional normed space and $\mathcal{M}$ is a smooth manifold. We assume that there is a reduced manifold of $(E_\varepsilon)$ given by the graph of a function $\phi\colon \mathcal{M}\to Y$ and satisfying an appropriate hyperbolicity assumption with unstable dimension $k\in{\mathbb N}_0$. We prove that every Morse decomposition $(M_p)_{p\in P}$ of a compact isolated invariant set $S_0$ of the reduced equation $$ \dot x=h(\phi(x),x,0) $$ gives rises, for $\varepsilon> 0$ small, to a Morse decomposition $(M_{p,\varepsilon})_{p\in P}$ of an isolated invariant set $S_\varepsilon$ of $(E_\varepsilon)$ such that $(S_\varepsilon,(M_{p,\varepsilon})_{p\in P})$ is close to $(\{0\}\times S_0,(\{0\}\times M_p)_{p\in P})$ and the (co)homology index braid of $(S_\varepsilon,(M_{p,\varepsilon})_{p\in P})$ is isomorphic to the (co)homology index braid of $(S_0,(M_{p})_{p\in P})$ shifted by $k$ to the left.