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

In this paper we consider a piecewise smooth $2$-dimensional system $$\dot{\vec{x}}=\vec{g}(\vec{x})+\epsilon \vec{g}(t,\vec{x},\epsilon )$$ where $\epsilon >0$ is a small parameter and $\vec{f}$ is discontinuous along a curve ${\mathrm{\Omega }}^0$. We assume that $\vec{0}$ is a critical point for any $\epsilon \ge 0$, and that for $\epsilon =0$ the system admits a trajectory $\vec{\gamma }(t)$ homoclinic to $\vec{0}$ and crossing transversely ${\mathrm{\Omega }}^0$ in $\vec{\gamma }(0)$. 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 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 $\vec{g}$ 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 $\mathrm{\Sigma }$ from which the chaotic pattern originates. The results are illustrated by examples.