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

• Lei Wang ^[1] ; Xiaoqian Zhang ^[1] ; Xu Li ^[1]
  1. [1] Hefei University

     Hefei University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 5, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • 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.

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