Two Limit Cycles in Liénard Piecewise Linear Differential Systems

Abstract

Some techniques for studying the existence of limit cycles for smooth differential systems are extended to continuous piecewise-linear differential systems. Rigorous new results are provided on the existence of two limit cycles surrounding the equilibrium point at the origin for systems with three zones separated by two parallel straight lines without symmetry.

Limit Cycles for Piecewise Linear Differential Systems via Poincaré–Miranda Theorem

“…Yang gave numerical evidences of a piecewise linear system with two zones and a discontinuity straight line, having three nested limit cycles ( [3]). A proof based on the Newton-Kantorovich theorem of the existence of these limit cycles for this example and a nearby one, was given by J. Llibre and E. Ponce ( [5]). A different proof, from a bifurcation viewpoint, was presented by E. Freire, E. Ponce and F. Torres in [1].…”

“…In this work we present a new example, again with 3 limit cycles, inspired on the ones given in [3,5]. The main contribution is that our proof relies on the so called Poincaré-Miranda theorem and it is very simple.…”

Global studies on a continuous planar piecewise linear differential system with three zones

“…In recent years, there has been tremendous interest in developing piecewise linear differential systems, see [2,4,5,6,7,8,9,10,11,15,16,17,20,21,22,23,24,26,27,30,33] and references therein. In applied science and engineering, piecewise linear differential systems can model a large number of nonlinear problems.…”

“…In applied science and engineering, piecewise linear differential systems can model a large number of nonlinear problems. Particularly, the global dynamics of some nonlinear models can be approximated by piecewise linear differential systems, such as, some memristor oscillators (see [1,4,5,8,9,23,24]) and FitzHugh-Nagumo system (see [28,29,31]). Although piecewise linear differential systems seem simple, they have rich and complex dynamics even with the low-dimensional spaces.…”

Integral characterization for Poincaré half-maps in planar linear systems

“…These two deficiencies are even further important if we take into account that almost every works about Poincaré maps for piecewise linear systems use direct integration of the systems, what is accompanied by large case-by-case studies. The valuable works [1,9,21,22,20], sorted by year of publication, are a few examples of this case-by-case studies from the early years to nowadays. Moreover, each one of these different cases requires individual techniques.…”