Periodic orbits in hysteretic systems with real eigenvalues

Abstract

Planar piecewise linear systems with hysteresis coming from a dimensional reduction in symmetric 3D systems with slow–fast dynamics are considered. We concentrate our attention on the cases of saddle and node dynamics, determining the existence and stability of periodic orbits as well as possible bifurcations. Our analysis rigorously shows, apart from standard bifurcations, as saddle-node bifurcation of periodic orbits and homoclinic and heteroclinic connections, the existence of specific grazing bifurcation. A pitchfork bifurcation of periodic orbits has been also detected, being responsible for the coexistence of up to four different periodic orbits. We illustrate the usefulness of the achieved theoretical results by justifying the appearance of periodic orbits in a concrete 3D system.