Geometric singular perturbation theory for non-smooth dynamical systems

Abstract

In this article we deal with singularly perturbed Filippov systems $Z_{\varepsilon}$:

(1) \dot{x} = \begin{cases} F(x,y,\varepsilon) &\text{if } h(x,y,\varepsilon) \leq 0,

G(x,y,\varepsilon) &\text{if } h(x,y,\varepsilon) \geq 0,

\end{cases} \quad \varepsilon \dot{y} = H(x,y,\varepsilon)

where $\varepsilon \in \mathbb{R}$ is a small parameter, $x \in \mathbb{R}^n$, $n\geq2,$ and $y \in \mathbb{R}$ denote the slow and fast variables, respectively, and $F$, $G$, $h$, and $H$ are smooth maps. We study the effect of singular perturbations at typical singularities of $Z_0$. Special attention will be dedicated to those points satisfying $q \in \{h(x,y,0) = 0\} \cap \{H(x,y,0) = 0\}$ where $F$ or $G$ is tangent to $\{h(x,y,0) = 0\}$. The persistence and the stability properties of those objects are investigated.