Limit Cycles in the Discontinuous Planar Piecewise Linear Systems with Three Zones

• Li, Zhengkang ^[1] ; Liu, Xingbo ^[1]
  1. [1] East China Normal University

     East China Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 3, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00496-4
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we investigate the existence of limit cycles for the discontinuous planar piecewise linear systems with three zones separated by two parallel straight lines. Based on the methods of first integral and Poincaré map, we present the maximum number of limit cycles in the normal forms of systems with boundary focus-center-boundary focus and boundary focus-center-center types, respectively. Then we show that such discontinuous piecewise linear systems can have at most three limit cycles, being two of them of four intersection points type and the third one of two intersection points type.

• Referencias bibliográficas
  • 1. Andronov, A.A., Vitt, A., Khaikin, S.: Theory of Oscillations. Pergamon Press, Oxford (1966)
  • 2. Battelli, F., Feckan, M.: On the Poincaré–Adronov–Melnikov method for the existence of grazing impact periodic solutions of differential...
  • 3. Braga, D.C., Mello, L.F.: More than three limit cycles in discontinuous piecewise linear differential systems with two zones in the plane....
  • 4. Buzzi, C., Pessoa, C., Torregrosa, J.: Piecewise linear perturbations of a linear center. Discrete Contin. Dyn. Syst. 33, 3915–3936 (2013)
  • 5. Chen, X., Llibre, J., Zhang, W.: Averaging approach to cyclicity of hopf bifurcation in planar linearquadratic polynomial discontinuous...
  • 6. Corinto, F., Ascoli, A., Gilli, M.: Nonlinear dynamics of memristor oscillators. IEEE Trans. Circuits Syst. I. Regul. Pap. 58, 1323–1336...
  • 7. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Applied Mathematical...
  • 8. Euzébio, R.D., Llibre, J.: On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated...
  • 9. Euzébio, R.D., Pazim, R., Ponce, E.: Jump bifurcations in some degenerate planar piecewise linear differential systems with three zones....
  • 10. Fonseca, A.F.D., Llibre, J., Mello, L.F.: Limit cycles in planar piecewise linear hamiltonian systems with three zones without equilibrium...
  • 11. Freire, E., Ponce, E., Rodrigo, F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with two zones. Int. J. Bifur....
  • 12. Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11, 181–211 (2012)
  • 13. Freire, E., Ponce, E., Torres, F.: A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear...
  • 14. Gasull, A., Torregrosa, J., Zhang, X.: Piecewise linear differential systems with an algebraic line of separation. Electron. J. Differ....
  • 15. Giannakopoulos, F., Pliete, K.: Planar systems of piecewise linear differential equations with a line of discontinuity. Nonlinearity 14,...
  • 16. Han, M., Sheng, L., Zhang, X.: Bifurcation theory for finitely smooth planar autonomous differential systems. J. Differ. Equ. 264, 3596–3618...
  • 17. Han, M., Zhang, W.: On Hopf bifurcation in non-smooth planar systems. J. Differ. Equ. 248, 2399–2416 (2010)
  • 18. Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902)
  • 19. Huan, S., Yang, X.: On the number of limit cycles in general planar piecewise linear systems. Discrete Contin. Dyn. Syst. A 32, 2147–2164...
  • 20. Huan, S., Yang, X.: Limit cycles in a family of planar piecewise linear differential systems with a nonregular separation line. Int. J....
  • 21. Huan, S.: On the number of limit cycles in general planar piecewise linear differential systems with two zones having two real equilibria....
  • 22. Jimenez, J., Llibre, J., Medrado, J.: Crossing limit cycles for piecewise linear differential centers separated by a reducible cubic curve....
  • 23. Kuznetsov, Yu.A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems. Int. J. Bifur. Chaos Appl. Sci. Eng....
  • 24. Li, L.: Three crossing limit cycles in planar piecewise linear systems with saddle-focus type. Electron. J. Qual. Theory Differ. Equ....
  • 25. Li, S., Cen, X., Zhao, Y.: Bifurcation of limit cycles by perturbing piecewise smooth integrable nonhamiltonian systems. Nonlinear Anal....
  • 26. Llibre, J., Novaes, D.D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn....
  • 27. Llibre, J., Ponce, E.: Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discrete...
  • 28. Llibre, J., Ponce, E., Valls, C.: Uniqueness and non-uniqueness of limit cycles for piecewise linear differential systems with three zones...
  • 29. Llibre, J., Ponce, E., Valls, C.: Two limit cycles in Liénard piecewise linear differential systems. J. Nonlinear Sci. 29, 1499–1522 (2019)
  • 30. Llibre, J., Teruel, A.E.: Introduction to the Qualitative Theory of Differential Systems, Planar, Symmetric and Continuous Piecewise Linear...
  • 31. Llibre, J., Teixeira, M.A.: Piecewise linear differential systems with only centers can create limit cycles? Nonlinear Dyn. 91, 249–255...
  • 32. Llibre, J., Zhang, X.: Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and...
  • 33. Llibre, J., Zhang, X.: Limit cycles created by piecewise linear centers. Chaos 29, (2019)
  • 34. Llibre, J., Zhang, X.: Limit cycles for discontinuous planar piecewise linear differential systems separated by an algebraic curve. Int....
  • 35. Novaes, D.D., Ponce, E.: A simple solution to the Braga–Mello conjecture. Int. J. Bifurc. Chaos Appl. Sci. Eng. 25, 1283–1293 (2015)
  • 36. Poincaré, H.: Sur L’intégration algébrique des équations différentielles du premier ordre et du premier degré I. Rendiconti del Circolo...
  • 37. Poincaré, H.: Sur L’intégration algébrique des équations différentielles du premier ordre et du premier degré II. Rendiconti del Circolo...
  • 38. Ponce, Elisabet, Ros, J., Vela, E.: The boundary focus-saddle bifurcation in planar piecewise linear systems. Application to the analysis...
  • 39. Shafarevich, I.R.: Basic Algebraic Geometry. Springer, New York (1974)
  • 40. Sheng, L., Han, M., Tian, Y.: On the number of limit cycles bifurcating from a compound polycycle. Int. J. Bifur. Chaos Appl. Sci. Eng....
  • 41. Simpson, D.J.W.: Bifurcations in Piecewise-Smooth, Continuous Systems. World Scientific Series on Nonlinear Science A, World Scientific,...
  • 42. Wang, J., He, S., Huang, L.: Limit cycles induced by threshold nonlinearity in planar piecewise linear systems of node-focus or node-center...
  • 43. Xiao, D., Xiong, Y., Han, M.: Limit cycle bifurcations by perturbing a quadratic integrable system with a triangle. J. Differ. Equ. 260,...
  • 44. Yu, P., Han, M., Li, J.: An improvement on the number of limit cycles bifurcating from a nondegenerate center of homogeneous polynomial...
  • 45. Zhao, Q., Yu, J.: Limit cycles of a class of discontinuous planar piecewise linear systems with three regions of Y-type. Qual. Theory....
  • 46. Zou, C., Liu, C., Yang, J.: On piecewise linear differential systems with n limit cycles of arbitrary multiplicities in two zones. Qual....