# Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

Jaume Llibre, Douglas D. Novaes, Marco A. Teixeira

Research output: Contribution to journalArticleResearchpeer-review

45 Citations (Scopus)

## Abstract

© 2015 World Scientific Publishing Company. We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N > 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N > 3.
Original language English 1550144 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 25 11 https://doi.org/10.1142/S0218127415501448 Published - 1 Oct 2015

## Keywords

• Discontinuous differential system
• limit cycle
• piecewise linear differential system

## Fingerprint

Dive into the research topics of 'Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones'. Together they form a unique fingerprint.