Abstract
Discontinuous piecewise linear systems with two zones are considered. A general canonical form that includes all the possible configurations in planar linear systems is introduced and exploited. It is shown that the existence of a focus in one zone is sufficient to get three nested limit cycles, independently on the dynamics of the another linear zone. Perturbing a situation with only one hyperbolic limit cycle, two additional limit cycles are obtained by using an adequate parametric sector of the unfolding of a codimension-two focus-fold singularity.
Similar content being viewed by others
References
Andronov, A., Vitt, A., Khaikin, S.: Theory of Oscillations. Pergamon Press, Oxford (1966)
Artés, J.C., Llibre, J., Medrado, J.C., Teixeira, M.A.: Piecewise linear differential systems with two real saddles. Math. Comp. Sim. (2013). 95, 13–22 (2013)
Braga, D.C., Mello, L.F.: Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane. Nonlinear Dynamics 73, 1283–1288 (2013)
Braga, D.C., L.F, Mello: More than three limit cycles in discontinuous piecewise linear differential systems with two zones in the plane. Int. J. Bifurc. Chaos 24, 1450056-1–1450056-10 (2014)
Buzzi, C., Pessoa, M., Torregrosa, J.: Piecewise linear perturbations of a linear center. Discrete Continuous Dyn. Syst. 9, 3915–3936 (2013)
di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Appl. Math. Sci., vol 163. Springer-Verlag London Ltd., London (2008)
Freire, E., Ponce, E., Rodrigo, F., Torres, F.: Bifurcation sets of continuous piecewise linear systems with two zones. Int. J. Bifurc. Chaos 8, 2073–2097 (1998)
Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11, 181–211 (2012)
Freire, E., Ponce, E., Torres, F.: The discontinuous matching of two planar linear foci can have three nested crossing limit cycles. Publ. Mat. 221–253. doi:10.5565/PUBLMAT_Extra14_13 (2014)
Freire, E., Ponce, E., Torres, F.: On the critical crossing cycle bifurcation in planar filippov systems. (Preprint)
Guardia, M., Seara, T.M., Teixeira, M.A.: Generic bifurcations of low codimension of planar filippov systems. J. Differential Equations 250, 1967–2023 (2011)
Han, M., Zhang, W.: On Hopf bifurcation in non-smooth planar systems. J. Differential Equations 248, 2399–2416 (2010)
Huan, S., Yang, X.: On the number of limit cycles in general planar piecewise linear systems. Discrete Continuous Dyn. Syst. A 32, 2147–2164 (2012)
Huan, S., Yang, X.: On the number of limit cycles in general planar piecewise linear systems of node-node types. J. Math. Anal. Appl. 411, 340–353 (2013)
Huan, S., Yang, X.: Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics. Nonlinear Anal. 92, 82–95 (2013)
Llibre, J., Teixeira, M.A., Torregrosa, J.: Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation. Int. J. Bifurc. Chaos. 23, (2013). doi:10.1142/S0218127413500661
Llibre, J., Ponce, E.: Three nested limit cycles in discontinuous piecewise linear differential systems. Dyn. Continuous Discrete Impuls. Syst. B 19, 325–335 (2012)
Shui, S., Zhang, X., Li, J.: The qualitative analysis of a class of planar Filippov systems. Nonlinear Anal. 73, 1277–1288 (2010)
Simpson, D.J.W.: Bifurcations in Piecewise-Smooth Continuous Systems. World Scientific Series on Nonlinear Science A, vol 69. World Scientific, Singapore (2010)
Acknowledgments
Authors are partially supported by the Spanish Ministerio de Ciencia y Tecnologia, Plan Nacional I+D+I, in the frame of projects MTM2010-20907 and MTM2012-31821, and by the Consejería de Economía-Innovacíon-Ciencia-Empleo de la Junta de Andalucí a under grant P12-FQM-1658.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Freire, E., Ponce, E. & Torres, F. A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear Dyn 78, 251–263 (2014). https://doi.org/10.1007/s11071-014-1437-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-014-1437-7