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

Maria C. Carbinatto, Krzysztof P. Rybakowski

DOI: http://dx.doi.org/10.12775/TMNA.2008.046

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.

Keywords


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

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