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

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.

Keywords

• piecewise smooth vector field
• Hamiltonian system
• crossing limit cycles
• conics