Hopf bifurcation at infinity in 3D Relay systems

Abstract

A complete analysis of the limit cycle bifurcation from infinity in 3D Relay systems, which belong to the class of three-dimensional symmetric discontinuous piecewise linear systems with two zones, is presented. A criticality parameter is found, whose sign determines the character of the bifurcation. When such non-degeneracy parameter vanishes, a higher co-dimension bifurcation takes place, giving rise to the emergence of a curve of saddle–node bifurcations of periodic orbits, which allows to determine parameter regions where two limit cycles coexist. The existence of a large amplitude limit cycle that bifurcates from infinity is justified through a suitable adaptation of the closing equations method, and analytical expressions for its amplitude and period are provided. Derivatives of the corresponding transition maps are rigorously studied in order to characterize the stability of the bifurcating limit cycle. The theoretical results are applied to a specific family of 3D relay systems, where several high co-dimension bifurcation points are detected, organizing the bifurcation set of the family. © 2022 Elsevier B.V.