Limit Cycles from Planar Piecewise Linear Hamiltonian differential Systems with Two or Three Zones

TL;DR

This paper investigates the existence and maximum number of limit cycles in planar piecewise linear Hamiltonian systems with two or three zones, showing that two-zone systems have none and three-zone systems have at most one, with examples provided.

Contribution

It establishes the maximum number of limit cycles in such systems and provides explicit examples with a single limit cycle in three-zone cases.

Findings

Two-zone systems have no limit cycles.

Three-zone systems have at most one limit cycle.

Examples with exactly one limit cycle are constructed.

Abstract

In this paper, we study the existence of limit cycles in continuous and discontinuous planar piecewise linear Hamiltonian differential systems with two or three zones separated by straight lines and such that the linear systems that define the piecewise one have isolated singular points, i.e. centers or saddles. In this case, we show that if the planar piecewise linear Hamiltonian differential system is either continuous or discontinuous with two zones, then it has no limit cycles. Now, if the planar piecewise linear Hamiltonian differential system is discontinuous with three zones, then it has at most one limit cycle, and there are examples with one limit cycle. More precisely, without taking into account the position of the singular points in the zones, we present examples with the unique limit cycle for all possible combinations of saddles and centers.