Existence, Number and Stability of Periodic Orbits Induced by Homoclinic Loops in Three-Dimensional Piecewise Linear Systems with an Admissible Saddle-Focus

Abstract

For a class of three-dimensional piecewise linear systems with an admissible saddle-focus, the existence of three kinds of homoclinic loops is shown. Moreover, the birth and number of the periodic orbits induced by homoclinic bifurcation are investigated, and various sufficient conditions are obtained to guarantee the appearance of only one periodic orbit, finitely many periodic orbits or countably infinitely many periodic orbits. Furthermore, the stability of these newborn periodic orbits is analyzed in detail and some conclusions are made about them to be periodic saddle orbits or periodic sinks. Finally, some examples are given.