American Institute of Mathematical Sciences

Advanced
November  2016, 15(6): 2509-2526. doi: 10.3934/cpaa.2016047

On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line

Fangfang Jiang 1, , Junping Shi 2, , Qing-guo Wang 3, and  Jitao Sun 4,

1. 

School of Science, Jiangnan University, Wuxi, 214122, China
2. 

Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795
3. 

Institute for Intelligent Systems, the University of Johannesburg, South Africa
4. 

Department of Mathematics, Tongji University, Shanghai, 200092, China

Received  November 2015 Revised  May 2016 Published  September 2016

In this paper, we investigate the existence and uniqueness of crossing limit cycle for a planar nonlinear Liénard system which is discontinuous along a straight line (called a discontinuity line). By using the Poincaré mapping method and some analysis techniques, a criterion for the existence, uniqueness and stability of a crossing limit cycle in the discontinuous differential system is established. An application to Schnakenberg model of an autocatalytic chemical reaction is given to illustrate the effectiveness of our result. We also consider a class of discontinuous piecewise linear differential systems and give a necessary condition of the existence of crossing limit cycle, which can be used to prove the non-existence of crossing limit cycle.
Keywords: Liénard system, uniqueness., crossing limit cycle, (non) existence, Discontinuous system.
Mathematics Subject Classification: Primary: 34A36, 34C05; Secondary: 37N9.
Citation: Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
References:
[1]

Y. L. An and M. A. Han, On the number of limit cycles near a homoclinic loop with a nilpotent singular point,, \emph{J. Differential Equations}, 258 (2015), 3194.  doi: 10.1016/j.jde.2015.01.006.  Google Scholar

[2]

J. C. Artés, J. Llibre, J. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles,, \emph{Math. Comput. Simulation}, 95 (2014), 13.  doi: 10.1016/j.matcom.2013.02.007.  Google Scholar

[3]

V. Carmona, S. Fernández-García, E. Freire and F. Torres, Melnikov theory for a class of planar hybrid systems,, \emph{Phys. D}, 248 (2013), 44.  doi: 10.1016/j.physd.2013.01.002.  Google Scholar

[4]

Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems,, \emph{Nonlinear Anal.}, 69 (2008), 3610.  doi: 10.1016/j.na.2007.09.045.  Google Scholar

[5]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and its Applications (Soviet Series),, Kluwer Academic Publishers Group, (1988).  doi: 10.1007/978-94-015-7793-9.  Google Scholar

[6]

N. Forcadel, A. Ghorbel and S. Walha, Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model,, \emph{J. Dynam. Differential Equations}, 26 (2014), 1133.  doi: 10.1007/s10884-014-9403-0.  Google Scholar

[7]

E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems,, \emph{SIAM J. Appl. Dyn. Syst.}, 11 (2012), 181.  doi: 10.1137/11083928X.  Google Scholar

[8]

Z. Y. Hou and S. Baigent, Global stability and repulsion in autonomous kolmogorov systems,, \emph{Commun. Pure Appl. Anal}, 14 ().  doi: 10.3934/cpaa.2015.14.1205.  Google Scholar

[9]

S. Huan and X. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics,, \emph{Nonlinear Anal.}, 92 (2013), 82.  doi: 10.1016/j.na.2013.06.017.  Google Scholar

[10]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types,, \emph{J. Math. Anal. Appl.}, 411 (2014), 340.  doi: 10.1016/j.jmaa.2013.08.064.  Google Scholar

[11]

T. W. Hwang and H. J. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions,, \emph{J. Phys. A}, 38 (2005), 8211.  doi: 10.1088/0305-4470/38/38/003.  Google Scholar

[12]

I. D. Iliev, C. Z. Li and J. Yu, Bifurcations of limit cycles in reversible quadratic system with a center, a saddle and two nodes,, \emph{Commun. Pure Appl. Anal}, 9 (2010), 583.  doi: 10.3934/cpaa.2010.9.583.  Google Scholar

[13]

F. Jiang and J. Sun, On the uniqueness of limit cycles in discontinuous Liénard-type systems,, \emph{Electron. J. Qual. Theory Differ. Equ.}, (): 1.   Google Scholar

[14]

F. Jiang, J. Shi and J. Sun, On the number of limit cycles for discontinuous generalized linéard polynomial differential systems,, \emph{Int. J. Bifurcat. Chaos}, 25 (2015).  doi: 10.1142/S021812741550131X.  Google Scholar

[15]

F. Jiang and J. Sun, Existence and uniqueness of limit cycle in discontinuous planar differential systems,, \emph{Qual. Theor. Dyn. Syst.}, (): 1.  doi: 10.1007/s12346-015-0141-4.  Google Scholar

[16]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems,, \emph{Math. Biosci.}, 88 (1988), 67.  doi: 10.1016/0025-5564(88)90049-1.  Google Scholar

[17]

C. Z. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four,, \emph{J. Differential Equations}, 252 (2012), 3142.  doi: 10.1016/j.jde.2011.11.002.  Google Scholar

[18]

P. Liu, J. P. Shi, Y. W. Wang and X. H. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model,, \emph{J. Math. Chem.}, 51 (2013), 2001.  doi: 10.1007/s10910-013-0196-x.  Google Scholar

[19]

J. Llibre and A. C. Mereu, Limit cycles for discontinuous generalized Liénard polynomial differential equations,, \emph{Electron. J. Differential Equations}, ().   Google Scholar

[20]

J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree,, \emph{Nonlinearity}, 27 (2014), 563.  doi: 10.1088/0951-7715/27/3/563.  Google Scholar

[21]

J. Llibre, M. Ordó nez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry,, \emph{Nonlinear Anal. Real World Appl.}, 14 (2013), 2002.  doi: 10.1016/j.nonrwa.2013.02.004.  Google Scholar

[22]

J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities,, \emph{Nonlinearity}, 21 (2008), 2121.  doi: 10.1088/0951-7715/21/9/013.  Google Scholar

[23]

A. C. J. Luo, Discontinuous Dynamical Systems,, Higher Education Press, (2012).  doi: 10.1007/978-3-642-22461-4.  Google Scholar

[24]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour,, \emph{J. Theoret. Biol.}, 81 (1979), 389.  doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[25]

J. F. Wang, J. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey,, \emph{J. Math. Biol.}, 62 (2011), 291.  doi: 10.1007/s00285-010-0332-1.  Google Scholar

[26]

D. M. Xiao and Z. F. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey systems,, \emph{Nonlinearity}, 16 (2003), 1185.  doi: 10.1088/0951-7715/16/3/321.  Google Scholar

[27]

Y. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang, M.-S. Wang and X.-A. Yang, Theory of Limit Cycles, vol. 66 of Translations of Mathematical Monographs,, 2nd edition, ().   Google Scholar

[28]

Z. F. Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations,, \emph{Appl. Anal.}, 23 (1986), 63.  doi: 10.1080/00036818608839631.  Google Scholar

[29]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z.-X. Dong, Qualitative theory of differential equations, vol. 101 of Translations of Mathematical Monographs,, American Mathematical Society, (1992).   Google Scholar

show all references

References:
[1]

Y. L. An and M. A. Han, On the number of limit cycles near a homoclinic loop with a nilpotent singular point,, \emph{J. Differential Equations}, 258 (2015), 3194.  doi: 10.1016/j.jde.2015.01.006.  Google Scholar

[2]

J. C. Artés, J. Llibre, J. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles,, \emph{Math. Comput. Simulation}, 95 (2014), 13.  doi: 10.1016/j.matcom.2013.02.007.  Google Scholar

[3]

V. Carmona, S. Fernández-García, E. Freire and F. Torres, Melnikov theory for a class of planar hybrid systems,, \emph{Phys. D}, 248 (2013), 44.  doi: 10.1016/j.physd.2013.01.002.  Google Scholar

[4]

Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems,, \emph{Nonlinear Anal.}, 69 (2008), 3610.  doi: 10.1016/j.na.2007.09.045.  Google Scholar

[5]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and its Applications (Soviet Series),, Kluwer Academic Publishers Group, (1988).  doi: 10.1007/978-94-015-7793-9.  Google Scholar

[6]

N. Forcadel, A. Ghorbel and S. Walha, Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model,, \emph{J. Dynam. Differential Equations}, 26 (2014), 1133.  doi: 10.1007/s10884-014-9403-0.  Google Scholar

[7]

E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems,, \emph{SIAM J. Appl. Dyn. Syst.}, 11 (2012), 181.  doi: 10.1137/11083928X.  Google Scholar

[8]

Z. Y. Hou and S. Baigent, Global stability and repulsion in autonomous kolmogorov systems,, \emph{Commun. Pure Appl. Anal}, 14 ().  doi: 10.3934/cpaa.2015.14.1205.  Google Scholar

[9]

S. Huan and X. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics,, \emph{Nonlinear Anal.}, 92 (2013), 82.  doi: 10.1016/j.na.2013.06.017.  Google Scholar

[10]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types,, \emph{J. Math. Anal. Appl.}, 411 (2014), 340.  doi: 10.1016/j.jmaa.2013.08.064.  Google Scholar

[11]

T. W. Hwang and H. J. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions,, \emph{J. Phys. A}, 38 (2005), 8211.  doi: 10.1088/0305-4470/38/38/003.  Google Scholar

[12]

I. D. Iliev, C. Z. Li and J. Yu, Bifurcations of limit cycles in reversible quadratic system with a center, a saddle and two nodes,, \emph{Commun. Pure Appl. Anal}, 9 (2010), 583.  doi: 10.3934/cpaa.2010.9.583.  Google Scholar

[13]

F. Jiang and J. Sun, On the uniqueness of limit cycles in discontinuous Liénard-type systems,, \emph{Electron. J. Qual. Theory Differ. Equ.}, (): 1.   Google Scholar

[14]

F. Jiang, J. Shi and J. Sun, On the number of limit cycles for discontinuous generalized linéard polynomial differential systems,, \emph{Int. J. Bifurcat. Chaos}, 25 (2015).  doi: 10.1142/S021812741550131X.  Google Scholar

[15]

F. Jiang and J. Sun, Existence and uniqueness of limit cycle in discontinuous planar differential systems,, \emph{Qual. Theor. Dyn. Syst.}, (): 1.  doi: 10.1007/s12346-015-0141-4.  Google Scholar

[16]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems,, \emph{Math. Biosci.}, 88 (1988), 67.  doi: 10.1016/0025-5564(88)90049-1.  Google Scholar

[17]

C. Z. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four,, \emph{J. Differential Equations}, 252 (2012), 3142.  doi: 10.1016/j.jde.2011.11.002.  Google Scholar

[18]

P. Liu, J. P. Shi, Y. W. Wang and X. H. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model,, \emph{J. Math. Chem.}, 51 (2013), 2001.  doi: 10.1007/s10910-013-0196-x.  Google Scholar

[19]

J. Llibre and A. C. Mereu, Limit cycles for discontinuous generalized Liénard polynomial differential equations,, \emph{Electron. J. Differential Equations}, ().   Google Scholar

[20]

J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree,, \emph{Nonlinearity}, 27 (2014), 563.  doi: 10.1088/0951-7715/27/3/563.  Google Scholar

[21]

J. Llibre, M. Ordó nez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry,, \emph{Nonlinear Anal. Real World Appl.}, 14 (2013), 2002.  doi: 10.1016/j.nonrwa.2013.02.004.  Google Scholar

[22]

J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities,, \emph{Nonlinearity}, 21 (2008), 2121.  doi: 10.1088/0951-7715/21/9/013.  Google Scholar

[23]

A. C. J. Luo, Discontinuous Dynamical Systems,, Higher Education Press, (2012).  doi: 10.1007/978-3-642-22461-4.  Google Scholar

[24]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour,, \emph{J. Theoret. Biol.}, 81 (1979), 389.  doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[25]

J. F. Wang, J. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey,, \emph{J. Math. Biol.}, 62 (2011), 291.  doi: 10.1007/s00285-010-0332-1.  Google Scholar

[26]

D. M. Xiao and Z. F. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey systems,, \emph{Nonlinearity}, 16 (2003), 1185.  doi: 10.1088/0951-7715/16/3/321.  Google Scholar

[27]

Y. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang, M.-S. Wang and X.-A. Yang, Theory of Limit Cycles, vol. 66 of Translations of Mathematical Monographs,, 2nd edition, ().   Google Scholar

[28]

Z. F. Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations,, \emph{Appl. Anal.}, 23 (1986), 63.  doi: 10.1080/00036818608839631.  Google Scholar

[29]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z.-X. Dong, Qualitative theory of differential equations, vol. 101 of Translations of Mathematical Monographs,, American Mathematical Society, (1992).   Google Scholar

[1]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[2]

Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. doi: 10.3934/dcdsb.2019187
[3]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893
[4]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043
[5]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[6]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213
[7]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
[8]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[9]

Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
[10]

Hany A. Hosham, Eman D Abou Elela. Discontinuous phenomena in bioreactor system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2955-2969. doi: 10.3934/dcdsb.2018294
[11]

Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291
[12]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[13]

Peter Markowich, Jesús Sierra. Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. Kinetic & Related Models, 2019, 12 (2) : 347-356. doi: 10.3934/krm.2019015
[14]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557
[15]

Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137
[16]

Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 441-448. doi: 10.3934/dcds.2007.17.441
[17]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[18]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[19]

Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192
[20]

Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences