Homoclinic Bifurcations in a Class of Three-Dimensional Symmetric Piecewise Affine Systems

Many physical and engineering systems have certain symmetric properties. Homoclinic orbits play an important role in studying the global dynamics of dynamical systems. This paper focuses on the existence and bifurcations of homoclinic orbits to a saddle in a class of three-dimensional one-parameter three-zone symmetric piecewise affine systems. Based on the analysis of the Poincaré maps, the systems have two types of limit cycles and do not have chaotic invariant sets near the homoclinic orbits. In addition, the paper provides a constant [Formula: see text] to study the homoclinic bifurcations to limit cycles for the case [Formula: see text]. Two examples with simulations of the homoclinic orbits and the limit cycles are given to illustrate the effectiveness of the results.