Three crossing limit cycles in planar piecewise linear systems with saddle-focus type

Liping Li

doi:10.14232/ejqtde.2014.1.70

Abstract

This paper presents an analysis on the appearance of limit cycles in planar Filippov system with two linear subsystems separated by a straight line. Under the restriction that the orbits with points in the sliding and escaping regions are not considered, we provide firstly a topologically equivalent canonical form of saddle-focus dynamic with five parameters by using some convenient transformations of variables and parameters. Then, based on a very available fourth-order series expansion of the return map near an invisible parabolic type tangency point, we show that three crossing limit cycles surrounding the sliding set can be bifurcated from generic codimensionthree singularities of planar discontinuous saddle-focus system. Our work improves and extends some existing results of other researchers.

Highlights

Piecewise linear systems often appear in the descriptions of many real processes such as dry friction in mechanical systems or switches in electronic circuits; for instance, see [3, 5, 7, 19, 20]. This kind of systems are generally modeled by ordinary differential equations with discontinuous right-hand sides which can exhibit very complicated dynamics and rich bifurcation phenomenons
One of the main problems in qualitative theory of piecewise linear systems is the determination of limit cycles
The occurrence of limit cycle in smooth systems can be provided through the analysis of Hopf bifurcation surrounding a singular point, such an approach fails for piecewise linear systems since the basic requirement of smoothness is not fulfilled here

Summary

Introduction

Piecewise linear systems often appear in the descriptions of many real processes such as dry friction in mechanical systems or switches in electronic circuits; for instance, see [3, 5, 7, 19, 20]. The occurrence of limit cycle in smooth systems can be provided through the analysis of Hopf bifurcation surrounding a singular point, such an approach fails for piecewise linear systems since the basic requirement of smoothness is not fulfilled here For this reason, many authors have contributed to develop several valid systematic methods in order to overcome the obstacles in recent years; for instance, see [1, 2, 6, 9, 11,12,13,14, 16,17,18, 22]. This implies that the conjecture in [12] is not correct

Preliminary definitions and notations

Canonical form of the saddle-focus dynamics

Limit cycles in saddle-focus canonical form

Conclusion