Global dynamics of a class of planar piecewise linear discontinuous system with two zones

• Qian Tong ^[1] ; Shimin Li ^[1]
  1. [1] Hangzhou Normal University

     Hangzhou Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 6, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we investigate a class of planar piecewise linear discontinuous system defined in two zones separated by a straight line. First, we establish necessary conditions for the existence of both crossing and sliding limit cycles in such system. Subsequently, employing the Poincaré compactification technique, we conduct a thorough analysis of the infinite singular points. Finally, we derive the global phase portraits within the Poincaré ball for this class of system, considering five distinct types: C-C, N-N, -, F-F, and S-S configurations.

• Referencias bibliográficas
  • 1. Filippov, A.F.: “Differential equations with discontinuous right-hand sides," Math. Appl. (Sov.Ser.) 18, Kluwer Academic Publishers...
  • 2. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-smooth dynamical systems: Theory and Applications. Springer, Appl....
  • 3. Leine, R.I., Nijmeijer, H.: “Dynamics and bifurcations of non-smooth mechanical systems," Springer- Verlag (2004)
  • 4. Zhusubaliyev, Z.T., Mosekilde, E.P:“Bifurcations and chaos in piecewise-smooth dynamical systems," World Scientific (2003)
  • 5. Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11, 181–211 (2012)
  • 6. Freire, E., Ponce, E., Rodrigo, F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with two zones. Int. J. Bifur....
  • 7. Li, S., Llibre, J.: Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line. J....
  • 8. Li, T., Chen, H., Chen, X.: Crossing periodic orbits of nonsmooth Liénard systems and applications. Nonlinearity 33(11), 5817–5838 (2020)
  • 9. Li, S., Liu, C., Llibre, J.: The planar discontinuous piecewise linear refracting systems have at most one limit cycle. Nonlinear Anal....
  • 10. Carmona, V., Fernández-Sánchez, F., Novaes, D.: Uniqueness and stability of limit cycles in planar piecewise linear differential systems...
  • 11. Li, S., Llibre, J.: Phase portraits of planar piecewise linear refracting systems: focus-saddle case. Nonlinear Anal. Real World Appl....
  • 12. Shao, Y., Li, S.,Wu, K.: Global phase portraits of planar piecewise linear refracting systems of saddlesaddle type. Nonlinear Anal. Real...
  • 13. Liang, H., Li, S., Zhang, X.: Limit cycles and global dynamics of planar piecewise linear refracting systems of focus-focus type. Nonlinear...
  • 14. Zhao, H.,Wu, K., Shao,Y.: Global dynamics of a planar piecewise linear refracting system of node-node types. Int. J. Bifur. Chaos 32(13),...
  • 15. Wang, Y., Wei, Z., Liu, H., Zhang, W.: Phase portraits of planar piecewise linear refracted systems: Node-saddle case. Discrete Contin....
  • 16. Shao, Y., Guan, H., Li, S., Fu, H.: The global dynamics of linear refracting systems of focus-node or center-node type. Nonlinear Anal....
  • 17. Huan, S., Yang, X.: On the number of limit cycles in general planar piecewise linear systems. Discrete Contin. Dyn. Syst. 32(6), 2147–2164...
  • 18. Llibre, J., Ponce, E.: Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discrete...
  • 19. Carmona, V., Sánchez, F.F., Novaes, D.D.: Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems...
  • 20. Tonnelier, A., Gerstner, W.: Piecewise linear differential equations and integrate-and-fire neurons: Insights from two-dimensional membrane...
  • 21. Li, T.: Limit cycles and bifurcations in a class of discontinuous piecewise linear systems. Int. J. Bifur. Chaos 29(10), 1950135 (2019)
  • 22. Guardia, M., Seara, T.M., Teixeira, M.A.: Generic bifurcations of low codimension of planar Filippov Systems. J. Diff. Equat. 250(4),...
  • 23. Freire, E., Ponce, E., Torres, F.: A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear...
  • 24. Zhang, Z.F.: Qualitative theory of differential equations. Amer. Math, SOC (1992)
  • 25. Huan, S.,Yang, X.: Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics. Nonlinear Anal. 92,...