Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones

Abstract

This work is devoted to study the existence of periodic solutions for a family of planar discontinuous differential systems Z(x, y; ε) with many zones. We show that for |ε| = 0 sufficiently small the averaged functions at any order control the existence of crossing limit cycles for systems in this family. We also provide some examples dealing with nonlinear centers when ε = 0.

Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold

“…, M, then the extra term f * 2 vanishes. Under one of these two last assumptions Theorem B provides the known results given in [3,20] for smooth systems, and in [15,18,19,23] for nonsmooth differential systems.…”

On the torus bifurcation in averaging theory

“…The next results were proved in [6]. Nonsmooth versions of these results can be found in [5]. Lemma 5 ([6]).…”

Bifurcations from families of periodic solutions in piecewise differential systems

“…In [22], it was proven that the average functions defined for smooth cases can be computed using Bell polynomials. In [17], the authors did the same for the non-smooth case.…”

“…and solving this linear differential equation we get the expression of w 1 1 (t, z) described in the statement of the lemma. For more details see [17].…”