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

#### Abstract

We consider the singularly perturbed system of ordinary differential equations

$$

\aligned

\varepsilon\dot y&=f(y,x,\varepsilon),

\\

\dot x&=h(y,x,\varepsilon)

\endaligned

\leqno(E_\varepsilon)

$$

on $Y\times \Cal{M}$, where $Y$ is a finite dimensional normed space and $\Cal{M}$ is a smooth manifold. We assume that there is

a reduced manifold of $(E_\varepsilon)$ given by the graph of a function $\phi\co \Cal{M}\to Y$

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.

$$

\aligned

\varepsilon\dot y&=f(y,x,\varepsilon),

\\

\dot x&=h(y,x,\varepsilon)

\endaligned

\leqno(E_\varepsilon)

$$

on $Y\times \Cal{M}$, where $Y$ is a finite dimensional normed space and $\Cal{M}$ is a smooth manifold. We assume that there is

a reduced manifold of $(E_\varepsilon)$ given by the graph of a function $\phi\co \Cal{M}\to Y$

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.

#### Keywords

Singular perturbations; differential equations on manifolds; Conley index; (co)homology index braid; continuation properties

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