Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold

TL;DR

This paper develops Melnikov functions for nonsmooth differential systems with a cubic switching manifold, establishing a lower bound for the Hilbert number and revealing new terms in second order analysis due to discontinuities.

Contribution

It generalizes Melnikov analysis to nonsmooth systems with nonlinear switching, providing new second order terms and a lower bound for the Hilbert number.

Findings

Lower bound of 7 for the Hilbert number in the studied family.

Second order Melnikov function includes a new term from discontinuity jumps.

First order Melnikov function remains unchanged from the smooth case.

Abstract

We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function.