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

#### 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.

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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