Resumen de Sliding Homoclinic Loops in Planar Piecewise Linear Systems: Exact Existence, Bifurcation Mechanisms and Global Dynamics

Lei Wang, Xiaoqian Zhang, Xu Li

• This paper investigates bifurcations and global dynamics induced by sliding homoclinic loops in planar piecewise-linear systems. Focusing on two fundamental classes (saddle-focus and saddle-center types), we first characterize tangent points, Poincaré maps, sliding sets, and pseudo-equilibria with their saddle-node bifurcations. Moreover, we establish the necessary and sufficient conditions for the existence of sliding homoclinic loops, proving a real unstable focus is essential. Furthermore, we determine the one-side stability and basins of attraction for the sliding homoclinic loops. Finally, we uncover novel bifurcation phenomena including co-occurrence of saddle-node and sliding homoclinic bifurcations, emergence of sliding limit cycles, and switching of the basins of attraction.