On the suspension isomorphism for index braids in a singular perturbation problem
Keywords
Singular perturbations, differential equations on manifolds, Conley index, (co)homology index braid, continuation propertiesAbstract
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.Downloads
Published
2008-12-01
How to Cite
1.
CARBINATTO, Maria C. and RYBAKOWSKI, Krzysztof P. On the suspension isomorphism for index braids in a singular perturbation problem. Topological Methods in Nonlinear Analysis. Online. 1 December 2008. Vol. 32, no. 2, pp. 199 - 225. [Accessed 26 April 2024].
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