Crossing Limit Cycles of Planar Piecewise Linear Hamiltonian Systems without Equilibrium Points

Rebiha Benterki,Jaume LLibre

doi:10.3390/math8050755

Abstract

In this paper, we study the existence of limit cycles of planar piecewise linear Hamiltonian systems without equilibrium points. Firstly, we prove that if these systems are separated by a parabola, they can have at most two crossing limit cycles, and if they are separated by a hyperbola or an ellipse, they can have at most three crossing limit cycles. Additionally, we prove that these upper bounds are reached. Secondly, we show that there is an example of two crossing limit cycles when these systems have four zones separated by three straight lines.

Highlights

IntroductionThe problem of the existence of limit cycles and mainly the problem of controlling their maximum number are two of the most difficult problems in the qualitative theory of differential systems in the plane
The problem of the existence of limit cycles and mainly the problem of controlling their maximum number are two of the most difficult problems in the qualitative theory of differential systems in the plane.We solve these two problems for the class of discontinuous piecewise differential systems here considered.We recall that a limit cycle is a periodic orbit of a differential system, which is isolated in the set of all periodic orbits of the system.Limit cycles appear in a natural way in many applications
We study the existence of crossing limit cycles of the planar piecewise linear

Summary

Introduction

The problem of the existence of limit cycles and mainly the problem of controlling their maximum number are two of the most difficult problems in the qualitative theory of differential systems in the plane. In 2015, Llibre et al [12] proved that if we separate the planar discontinuous piecewise linear differential centers by a straight line, we cannot have any limit cycle. Our first objective is to provide the exact maximum number of crossing limit cycles of planar discontinuous piecewise linear Hamiltonian systems without equilibrium points (PHS) and separated by a conic Σ. The second objective of this paper is to study the crossing limit cycles of piecewise smooth differential systems such that in each piece, the differential system is linear, Hamiltonian, and without equilibrium points. Continuous planar piecewise Hamiltonian systems without equilibrium points with four zones satisfying C have no crossing limit cycles. There are discontinuous planar piecewise Hamiltonian systems without equilibrium points with four zones satisfying C, exhibiting exactly two crossing limit cycles; see Figure 4.

Proof of Theorem 1

Proof of Theorems 2 and 3

Conclusions