Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line

TL;DR

This paper addresses the bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line, solving the center-focus problem and identifying maximum limit cycles in both continuous and discontinuous cases.

Contribution

It introduces new results on the maximum number of limit cycles and solves the cyclicity problem for a specific class of piecewise quadratic systems.

Findings

Maximum of 3 limit cycles in the continuous class.

Existence of 7 small-amplitude limit cycles in the discontinuous class.

Solution to the center-focus problem for systems with an invariant straight line.

Abstract

We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being $3$ such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of $7$ limit cycles of small amplitude.