Limit cycles from a monodromic infinity in planar piecewise linear systems

TL;DR

This paper analyzes planar piecewise linear systems with a focus on limit cycles emerging from a periodic orbit at infinity, introducing a new approach for stability and bifurcation analysis that reveals complex bifurcation phenomena.

Contribution

It introduces a simplified canonical form and a direct coordinate approach to analyze bifurcations at infinity, uncovering new degeneracies and limit cycle bifurcations.

Findings

Up to three limit cycles can bifurcate from the periodic orbit at infinity.

Degeneracies of co-dimension three in Hopf bifurcations at infinity are possible.

A new mechanism explains the maximum number of limit cycles in these systems.

Abstract

Planar piecewise linear systems with two linearity zones separated by a straight line and with a periodic orbit at infinity are considered. By using some changes of variables and parameters, a reduced canonical form with five parameters is obtained. Instead of the usual Bendixson transformation to work near infinity, a more direct approach is introduced by taking suitable coordinates for the crossing points of the possible periodic orbits with the separation straight line. The required computations to characterize the stability and bifurcations of the periodic orbit at infinity are much easier. It is shown that the Hopf bifurcation at infinity can have degeneracies of co-dimension three and, in particular, up to three limit cycles can bifurcate from the periodic orbit at infinity. This provides a new mechanism to explain the claimed maximum number of limit cycles in this family of systems. The centers at infinity classification together with the limit cycles bifurcating from them are also analyzed.