Abstract
In this paper, we investigate some qualitative properties of crossing limit cycles for a discontinuous symmetric Liénard system with two zones separated by a straight line. In each zone, it is a smooth Liénard system. Firstly, by Poincaré mapping method and geometrical analysis, we provide two criteria concerning the existence, uniqueness and stability of a crossing limit cycle. Secondly, we consider the position problem of the unique crossing limit cycle. Several lemmas are given to obtain an explicit upper bound of amplitude of the limit cycle. Finally, an application to van der Pol model with discontinuous vector field is given, and Matlab simulations are presented to illustrate the obtained theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Artés, J.C., Llibre, J., Medrado, J.C., Teixeira, M.A.: Piecewise linear differential systems with two real saddles. Math. Comput. Simul. 95, 13–22 (2014)
Braga, D.D.C., Mello, L.F.: More than three limit cycles in discontinuous piecewise linear differential systems with two zones in the plane. Int. J. Bifurcat. Chaos 24(04), 1450056 (2014)
Braga, D.D.C., Mello, L.F.: Arbitrary number of limit cycles for planar discontinuous piecewise linear differential systems with two zones. Electron. J. Differ. Equ. 2015, 1–12 (2015)
Carmona, V., Fernández-García, S., Freire, E., Torres, F.: Melnikov theory for a class of planar hybrid systems. Physica D 248, 44–54 (2013)
Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides, Volume 18 of Mathematics and Its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, Translated from the Russian (1988)
Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11(1), 181–211 (2012)
Giannakopoulos, F., Pliete, K.: Planar systems of piecewise linear differential equations with a line of discontinuity. Nonlinearity 14(6), 1611–1632 (2001)
Hsu, S.B., Hwang, T.W.: Uniqueness of limit cycles for a predator–prey system of holling and leslie type. Canad. Appl. Math. Quart. 6(2), 91–117 (1998)
Huan, S., Yang, X.: Existence of limit cycles in general planar piecewise linear systems of saddle–saddle dynamics. Nonlinear Anal. 92, 82–95 (2013)
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(1), 340–353 (2014)
Hwang, T.W., Tsai, H.J.: Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions. J. Phys. A 38(38), 8211–8223 (2005)
Jiang, F., Shi, J.P., Wang, Q.G., Sun, J.: On the existence and uniqueness of a limit cycle for a linard system with a discontinuity line. Commun. Pure Appl. Anal. 15(16), 2509–2526 (2016)
Jiang, F., Sun, J.: Existence and uniqueness of limit cycle in discontinuous planar differential systems. Qual. Theory Dyn. Syst. 15(1), 67–80 (2016)
Kuang, Y., Freedman, H.I.: Uniqueness of limit cycles in Gause-type models of predator–prey systems. Math. Biosci. 88(1), 67–84 (1988)
Li, J.B.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 13(1), 47–106 (2003)
Liu, P., Shi, J.P., Wang, Y.W., Feng, X.H.: Bifurcation analysis of reaction–diffusion Schnakenberg model. J. Math. Chem. 51(8), 2001–2019 (2013)
Liu, Y., Han, M., Romanovski, V.G.: Some bifurcation analysis in a family of nonsmooth Liénard systems. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 25(4), 155005516 (2015)
Llibre, J., Ponce, E., Torres, F.: On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities. Nonlinearity 21(9), 2121–2142 (2008)
Llibre, J., Ponce, E., Valls, C.: Uniqueness and non-uniqueness of limit cycles for piecewise linear differential systems with three zones and no symmetry. J. Nonlinear Sci. 25(4), 861–887 (2015)
Sun, Y., Liu, L., Wu, Y.: The existence and uniqueness of positive monotone solutions for a class of nonlinear schrödinger equations on infinite domains. J. Comput. Appl. Math. 321, 478–486 (2017)
Wang, J.F., Shi, J.P., Wei, J.J.: Predator–prey system with strong Allee effect in prey. J. Math. Biol. 62(3), 291–331 (2011)
Wei, L.J., Zhang, X.: Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete Contin. Dyn. Syst. 36(5), 2803–2825 (2016)
Xiao, D.M., Zhang, Z.F.: On the uniqueness and nonexistence of limit cycles for predator–prey systems. Nonlinearity 16(3), 1185–1201 (2003)
Yang, L.J., Zeng, X.W.: An upper bound for the amplitude of limit cycles in Liénard systems with symmetry. J. Differ. Equ. 258(8), 2701–2710 (2015)
Zhang, Z.F., Ding, T.R., Huang, W.Z., Dong, Z.-X.: Qualitative Theory of Differential Equations, Volume 101 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1992. Translated from the Chinese by Anthony Wing Kwok Leung
Acknowledgements
The authors would like to thank the referees for their careful checking and helpful comments, which have improved the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the National Natural Science Foundation of China (11701224), the Provincial Youth Foundation of JiangSu Province (BK20170168), the China Postdoctoral Science Foundation (2017M611685), and the Provincial Outstanding Youth Foundation of JiangSu Province (BK20160001).
Rights and permissions
About this article
Cite this article
Jiang, F., Ji, Z. & Wang, Y. Qualitative Analysis of Crossing Limit Cycles in a Class of Discontinuous Liénard Systems with Symmetry. Qual. Theory Dyn. Syst. 18, 85–105 (2019). https://doi.org/10.1007/s12346-018-0278-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12346-018-0278-z