Markus-Yamabe Conjecture Holds for Planar Continuous Piecewise Linear Systems with Multiple Parallel Switching Lines

• Yuhong Zhang ^[2] ; Xiao-Song Yang ^[1] ; Weisheng Huang ^[2] ; Yan Tang ^[2]
  1. [1] Huazhong University of Science and Technology

     Huazhong University of Science and Technology

     China

     
  2. [2] South-Central Minzu University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • The Markus–Yamabe conjecture provides a foundational criterion for the global asymptotic stability of equilibrium points in smooth dynamical systems. This paper investigates its validity within the framework of planar continuous piecewise linear systems. We focus on systems whose phase space is partitioned into n + 1 zones by n parallel switching lines, a structure commonly arising in models with multi-stage stiffness. The main result establishes that if every linear subsystem defined within each zone is asymptotically stable, meaning its Jacobian matrix is Hurwitz, and the overall vector field is continuous across all switching lines, then the system possesses a unique, globally asymptotically stable equilibrium point. This finding extends the validity of the Markus–Yamabe conjecture to a broader class of piecewise-linear systems with multiple parallel switching boundaries, thereby complementing existing results for single-switching-line systems and addressing the gap for multi-zone configurations.

    Aphysical model of a single-degree-of-freedom oscillatorwith multi-stage spring stiffness is presented to illustrate the practical applicability of the theoretical framework, accompanied by numerical simulations confirming the global attracting property.

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