On the Number of Limit Cycles in General Planar Piecewise Linear Differential Systems with Two Zones Having Two Real Equilibria

• Huan, Song-Mei ^[1]
  1. [1] Huazhong University of Science and Technology

     Huazhong University of Science and Technology

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 1, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-020-00441-x
• Enlaces
  • Texto completo
• Resumen

  • A general family of planar piecewise linear ODEs with two zones both having a real focus and separated by a straight line is considered. By analyzing the number of zero points of a new function related to the intersection points of the trajectories of the linear subsystems with the separation line, complete results on the existence and number of limit cycles are obtained. In particular, complete parameter regions for the existence of 1–2 limit cycles are provided with two concrete examples, which will be helpful in studying some kinds of discontinuity-induced bifurcations (i.e., DIBs). Based on the main results, it is obtained that the family of planar piecewise linear ODEs with focus–focus dynamics separated by a straight line can have 3 limit cycles if and only if one subsystem has a real focus and the other one has a virtual focus. Moreover, it is showed that five is the least value of the number of parameters needed in canonical forms of such systems.

• Referencias bibliográficas
  • 1. Andronov, A.A., Vitt, A., Khaikin, S.: Theory of Oscillators. Pergamon Press, Oxford, New York, Toronto (1966)
  • 2. Barbashin, E.A.: Introduction to the Theory of Stability. Noordhoff, Groningen (1970)
  • 3. Barnet, S., Cameron, R.G.: Introduction to Mathematical Control Theory. Oxford University Press, New York (1985)
  • 4. Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific, River-Edge (1998)
  • 5. Brogliato, B.: Nonsmooth Mechanics. Springer, New York (1999)
  • 6. Braga, D.D.C., Mello, L.F.: Limit cycles in a family of discontinuou piecewise linear differential systems with two zones in the plane....
  • 7. Braga, D.D.C., Mello, L.F.: More than three limit cycles in discontinuous piecewise linear differential systems with two zones in the plane....
  • 8. Buzzi, C., Pessoa, C., Torregrosa, J.: Piecewise linear perturbations of a linear center. Discrete Contin. Dyn. Syst. 33(9), 3915–3936...
  • 9. Carmona, V., Freire, E., Ponce, E., Torres, F.: On simplifying and classifying piecewise-linear systems. IEEE Trans. Circuits Syst. I Fund....
  • 10. Carmona, V., Freire, E., Ponce, E., Torres, F.: The continuous matching of two stable linear systems can be unstable. Discrete Contin....
  • 11. Castillo, J., Llibre, J., Verduzco, F.: The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems. Nonlinear...
  • 12. Cardin, P.T., Torregrosa, J.: Limit cycles in planar piecewise linear differential systems with nonregular separation line. Phys. D 337,...
  • 13. Coombes, S., Thul, R., Wedgwood, K.C.A.: Nonsmooth dynamics in spiking neuron models. Phys. D 241, 2042–2057 (2012)
  • 14. Dercole, F., Gragnani, S., Rinaldi, S.: Bifurcation analysis of piecewise smooth ecological models. Theor. Popul. Biol. 72(2), 197–213...
  • 15. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems Theory and Applications. Springer, London...
  • 16. Freire, E., Ponce, E., Rodrigo, F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with two zone. Int. J. Bifur....
  • 17. Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11(1), 181–211 (2012)
  • 18. Freire, E., Ponce, E., Torres, F.: The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. Publ....
  • 19. Freire, E., Ponce, E., Torres, F.: A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear...
  • 20. Henry, C.: Differential equations with discontinuous righthand side for planning procedure. J. Econ. Theory 4, 541–551 (1972)
  • 21. Huan, S.M., Yang, X.S.: Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems. Nonlin. Anal. 75,...
  • 22. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise systems. Discrete Contin. Dyn. Syst 32, 2147–2164 (2012)
  • 23. Huan, S.M., Li, Q.D., Yang, X.S.: Chaos in three-dimensional hybrid systems and design of chaos generators. Nonlinear Dyn. 69, 1915–1927...
  • 24. Huan, S.M., Yang, X.S.: Existence of limit cycles in general planar piecewise linear systems of saddlesaddle dynamics. Nonlinear Anal....
  • 25. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise linear systems of node-node types. J. Math. Anal. Appl....
  • 26. Huan, S.M., Yang, X.S.: Existence of chaotic invariant set in a class of 4-dimensional piecewise linear dynamical systems. Int. J. Bifur....
  • 27. Huan, S.M.: Existence of invariant cones in general 3-dim homogeneous piecewise linear differential systems with two zones. Int. J. Bifur....
  • 28. Huan, S.M., Yang, X.S.: Limit cycles in a family of planar piecewise linear differential systems with a nonregular separation line. Int....
  • 29. Huan, S. M., Wu, T. T., Wang, L.: Poincaré bifurcations induced by a non-regular point on the discontinuity boundary in a family of planar...
  • 30. Ito, T.: A Filippov solution of a system of differential equations with discontinuous right-hand sides. Econ. Lett. 4, 349–354 (1979)
  • 31. Krivan, V.: On the Gause predator–prey model with a refuge: a fresh look at the history. J. Theor. Biol. 274, 67–73 (2011)
  • 32. Li, S., Llibre, J.: On the limit cycles of planar discontinous piecewise linear differential systems with a unique equilibrium. Discrete...
  • 33. Li, S., Llibre, J.: Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line. J....
  • 34. Liang, F., Romanovski, V.G., Zhang, D.X.: Limit cycles in small perturbations of a planar piecewise linear Hamiltonian system with a non-regular...
  • 35. Llibre, J., Ponce, E.: Three nested limit cycles in discontinous piecewise linear differential systems. Dyn. Contin. Discrete Impuls....
  • 36. Llibre, J., Teixeira, M.A., Torregrosa, J.: Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential...
  • 37. Llibre, J., Novaes, D.D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn....
  • 38. Llibre, J., Teixeira, M.A.: Piecewise linear differential systems without equilibria produce limit cycles? Nonlinear Dyn. 88, 157–164...
  • 39. Llibre, J., Zhang, X.: Limit cycles for discontinuous planar piecewise linear differential systems separated by an algebraic curve. Int....
  • 40. Maggio, G.M., di Bernardo, M., Kennedy, M.P.: Nonsmooth bifurcations in a piecewise linear model of the Colpitts oscillator. IEEE Trans....
  • 41. Mereu, A.C., Oliveira, R., Rodrigues, C.A.B.: Limit cycles for a class of discontinuou piecewise generalized Kukles differential systems....
  • 42. Novaes, D.D., Ponce, E.: A simple solution to the Braga-Mello conjecture. Int. J. Bifur. Chaos Appl. Sci. Eng. 25(1), 1550009 (2015)
  • 43. Stoker, J.J.: Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience Publishers Inc, New York, NY (1950)
  • 44. Tonnelier, A., Gerstner, W.: Piecewise linear differential equations and integrate-and-fire neurons: insights from two-dimensional membrane...
  • 45. Weiss, D., Küpper, T., Hosham, H.A.: Invariant manifolds for nonsmooth systems with sliding mode. Math. Comput. Simul. 110, 15–32 (2015)
  • 46. Wu, T.T., Wang, L., Yang, X.S.: Chaos generator design with piecewise affine systems. Nonlinear Dyn. 84, 817–832 (2016)