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 continuous vector field on $\mathcal{M}$. This allows us to extend our previously obtained results from [M.C. Carbinatto and K.P. Rybakowski, On the suspension isomorphism for index braids in a singular perturbation problem, Topological Methods in Nonl. Analysis 32 (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.