Conley index and homology index braids in singular perturbation problems without uniqueness of solutions

Maria C. Carbinatto, Krzysztof P. Rybakowski

Abstract


We define the concept of a Conley index and a homology index braid class for
ordinary differential equations of the form
\begin{equation}
\dot x= F_1(x),
\tag{$E$}
\end{equation}
where $\mathcal{M}$ is a $C^2$-manifold and $F_1$ is the principal part of
a < i> continuous vector field< /i> on $\mathcal{M}$.
This allows us to extend our previously obtained results
from [M.C. Carbinatto and K.P. Rybakowski, < i> On the suspension isomorphism for index braids in a
singular perturbation problem< /i> , Topological Methods in Nonl. Analysis < b> 32< /b> (2008), 199-225] on singularly perturbed systems
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 Banach space
and $\mathcal{M}$ is a $C^2$-manifold, to the case where the vector field
in $(E_\varepsilon)$ is continuous, but not necessarily locally Lipschitzian.

Keywords


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

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