A new construction for Melnikov chaos in piecewise-smooth planar systems

We consider a piecewise-smooth $2$-dimensional system \begin{equation*}\label{e.ab} \dot{\x}=\f (\x)+\ep\g(t,\x,\ep) \end{equation*} where $\ep>0$ is a small parameter and $\f$ is discontinuous along a curve $\Om^0$. We assume that $\vec{0}$ is a critical point for any $\ep \ge 0$, and that for $\ep=0$ the system admits a trajectory $\ga(t)$ homoclinic to $\vec{0}$ and crossing transversely $\Om^0$ in $\ga(0)$. In a previous paper, we have shown that, also in an $n$-dimensional setting, the classical Melnikov condition is enough to guarantee the persistence of the homoclinic orbit to perturbations, but more recently we have found an open condition, a geometric obstruction which is not possible in the smooth case, which prevents chaos for $2$-dimensional systems when $\g$ is periodic in $t$. In this paper, we show that when this obstruction is removed we have chaos as in the smooth case. The proofs involve a new construction of the set from which the chaotic pattern originates. The results are illustrated by examples.