Ir al contenido

Documat

A Mechanism of Generating Two Limit Cycles in Planar Piecewise Linear Systems with a Subsystem Having No Equilibria

  • Linlin Zhao [1] ; Jiafu Wang [1] ; Lihong Huang [2]
    1. [1] Changsha University of Science and Technology

      Changsha University of Science and Technology

      China

    2. [2] Changsha University

      Changsha University

      China

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
  • Idioma: inglés
  • Enlaces
  • Resumen

    • The goal of this paper is to study the number and stability of crossing limit cycles for planar piecewise linear systems with one subsystem having no equilibria and the other one possessing a focus or a center. By the approach of Poincaré maps and displacement functions, it is shown that there are at least two nested crossing limit cycles in some parameter regions. Moreover, sufficient conditions are established to ensure the existence of exactly two nested limit cycles or a unique one. As applications, our results are successfully applied to detect the existence and stability of a limit cycle for a Morris-Lecar model.

  • Referencias bibliográficas
    • 1. Banerjee, S., Verghese, G.C.: Nonlinear Phenomena in Power Electronics: Bifurcations, Chaos, Control, And Applications. IEEE Press, New...
    • 2. di Bernardo, M.,Budd, C.J.,Champneys, A.R.,Kowalczyk, P.: Piecewise-SmoothDynamical Systems: Theory and Applications. Springer-Verlag,...
    • 3. Li, T., Chen, X.: Degenerate grazing-sliding bifurcations in planar Filippov systems. J. Differ. Equ. 269(12), 11396–11434 (2020)
    • 4. Yu, J., Li, J.: Global asymptotic stability in an interactive wild and sterile mosquito model. J. Differ. Equ. 269(7), 6193–6215 (2020)
    • 5. Huang, L.,Wang, J.:Models Described by Differential Equations with Discontinous Right Hand Sides and Their Dynamics. Science Press, Beijing...
    • 6. Liu, S., Han, M., Li, J.: Bifurcation methods of periodic orbits for piecewise smooth systems. J. Differ. Equ. 275, 204–233 (2021)
    • 7. Xu, W., Han, M.: Hopf bifurcation of limit cycles in some piecewise smooth Liénard systems. Int. J. Bifurc. Chaos 30(12), 2050249 (2020)
    • 8. Chen, H., Tang, Y., Xiao, D.: On the uniqueness of limit cycles in second-order oscillators. J. Differ. Equ. 370, 140–166 (2023)
    • 9. Chen, H., Tang, Y., Wang, Z.: The discontinuous limit case of an archetypal oscillator with a constant excitation and van der Pol damping....
    • 10. Zheng, B., Yu, J.: At most two periodic solutions for a switching mosquito population suppression model. J. Dyn. Differ. Equ. 35, 2997–3009...
    • 11. Feng, X., Liu,Y., Ruan, S.,Yu, J.: Periodic dynamics of a single species modelwith seasonalMichaelis- Menten type harvesting. J. Differ....
    • 12. Chen, H., Wang, L., Zhang, X.: A necessary and sufficient condition on algebraic limit cycles of a hybrid van der Pol-Rayleigh oscillator....
    • 13. Han, M., Liu, S.: Hopf bifurcation in a class of piecewise smooth near-Hamiltonian systems. Bull. Sci. Math. 195, 103471 (2024)
    • 14. Tonnelier, A., Gerstner, W.: Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane...
    • 15. Coombes, S.: Neuronal networks with gap junctions: a study of piecewise linear planar neuron models. SIAM J. Appl. Dyn. Syst. 7(3), 1101–1129...
    • 16. Thul, R., Coombes, S.: Understanding cardiac alternans: a piecewise linear modeling framework. Chaos 20(4), 045102 (2010)
    • 17. Huang, L., Wang, J.: Global dynamics of piecewise smooth systems with switches depending on both discrete times and status. SIAM J. Appl....
    • 18. Huang, L., Wang, J.: Sawtooth periodic solutions of stable-unstable switching systems with statedependent switches at fixed sampling times....
    • 19. Llibre, J., Teixeira, M.A., Torregrosa, J.: Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential...
    • 20. Llibre, J., Novas, D.D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn....
    • 21. Llibre, J., Zhang, X.: Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and...
    • 22. Llibre, J., Teixeira, M.A.: Piecewise linear differential systems with only centers can create limit cycles? Nonlinear Dyn. 91, 249–255...
    • 23. Li, S., Llibre, J.: Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line. J....
    • 24. Wu, F., Wang, J., Huang, L.: Bifurcation analysis of a planar piecewise linear system of focus-focus type. Int. J. Bifurc. Chaos 33(14),...
    • 25. Chen, L., Liu, C.: The discontinuous planar piecewise linear systems with two improper nodes have at most one limit cycle. Nonlin. Anal....
    • 26. Han, M., Zhang,W.:OnHopf bifurcation in non-smooth planar systems. J. Differ. Equ. 248, 2399–2416 (2010)
    • 27. Huan, S., Yang, X.: On the number of limit cycles in general planar piecewise linear systems. Discr. Contin. Dyn. Syst. 32(6), 2147–2164...
    • 28. Llibre, J., Ponce, E.: Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discr....
    • 29. Freire, E., Ponce, E., Torres, F.: The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. Publ....
    • 30. Freire, E., Ponce, E., Torres, F.: A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear...
    • 31. Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11(1), 181–211 (2012)
    • 32. Huan, S.,Yang, X.: Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics. Nonlinear Anal. 92,...
    • 33. Huan, S., Yang, X.: On the number of limit cycles in general planar piecewise linear systems of node-node types. J. Math. Anal. Appl....
    • 34. Wang, J., Chen, X., Huang, L.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math....
    • 35. Wang, J., Huang, C., Huang, L.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear...
    • 36. Wang, J., He, S., Huang, L.: Limit cycles induced by threshold nonlinearity in planar piecewise linear systems of node-focus or node-center...
    • 37. Llibre, J., Teixeira, M.A.: Piecewise linear differential systems without equilibria produce limit cycles? Nonlinear Dyn. 88, 157–164...
    • 38. Berbache, A.: Explicit non algebraic limit cycle for a discontinuous piecewise differential systems separated by one straight line and...
    • 39. Kuznetsov, Y.A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems. Int. J. Bifurc. Chaos 13(8), 2157–2188...
    • 40. Guardia, M., Seara, T.M., Teixeira, M.A.: Generic bifurcations of low codimension of planar Filippov systems. J. Differ. Equ. 250(4),...
    • 41. Llibre, J., Ponce, E., Torres, F.: On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities....
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno