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

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.
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]

J. Differential Equations, 258 (2015), 3194-3247. doi: 10.1016/j.jde.2015.01.006.  Google Scholar

[2]

Math. Comput. Simulation, 95 (2014), 13-22. doi: 10.1016/j.matcom.2013.02.007.  Google Scholar

[3]

Phys. D, 248 (2013), 44-54. doi: 10.1016/j.physd.2013.01.002.  Google Scholar

[4]

Nonlinear Anal., 69 (2008), 3610-3628. doi: 10.1016/j.na.2007.09.045.  Google Scholar

[5]

Kluwer Academic Publishers Group, Dordrecht, 1988, doi: 10.1007/978-94-015-7793-9.  Google Scholar

[6]

J. Dynam. Differential Equations, 26 (2014), 1133-1169. doi: 10.1007/s10884-014-9403-0.  Google Scholar

[7]

SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211. 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]

Nonlinear Anal., 92 (2013), 82-95. doi: 10.1016/j.na.2013.06.017.  Google Scholar

[10]

J. Math. Anal. Appl., 411 (2014), 340-353. doi: 10.1016/j.jmaa.2013.08.064.  Google Scholar

[11]

J. Phys. A, 38 (2005), 8211-8223. doi: 10.1088/0305-4470/38/38/003.  Google Scholar

[12]

Commun. Pure Appl. Anal, 9 (2010), 583-610. 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]

Int. J. Bifurcat. Chaos, 25 (2015), 1550131. 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]

Math. Biosci., 88 (1988), 67-84. doi: 10.1016/0025-5564(88)90049-1.  Google Scholar

[17]

J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.  Google Scholar

[18]

J. Math. Chem., 51 (2013), 2001-2019. 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]

Nonlinearity, 27 (2014), 563-583. doi: 10.1088/0951-7715/27/3/563.  Google Scholar

[21]

Nonlinear Anal. Real World Appl., 14 (2013), 2002-2012. doi: 10.1016/j.nonrwa.2013.02.004.  Google Scholar

[22]

Nonlinearity, 21 (2008), 2121-2142. doi: 10.1088/0951-7715/21/9/013.  Google Scholar

[23]

Higher Education Press, Beijing; Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-22461-4.  Google Scholar

[24]

J. Theoret. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[25]

J. Math. Biol., 62 (2011), 291-331. doi: 10.1007/s00285-010-0332-1.  Google Scholar

[26]

Nonlinearity, 16 (2003), 1185-1201. 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]

Appl. Anal., 23 (1986), 63-76. doi: 10.1080/00036818608839631.  Google Scholar

[29]

American Mathematical Society, Providence, RI, 1992, Google Scholar

show all references

References:
[1]

J. Differential Equations, 258 (2015), 3194-3247. doi: 10.1016/j.jde.2015.01.006.  Google Scholar

[2]

Math. Comput. Simulation, 95 (2014), 13-22. doi: 10.1016/j.matcom.2013.02.007.  Google Scholar

[3]

Phys. D, 248 (2013), 44-54. doi: 10.1016/j.physd.2013.01.002.  Google Scholar

[4]

Nonlinear Anal., 69 (2008), 3610-3628. doi: 10.1016/j.na.2007.09.045.  Google Scholar

[5]

Kluwer Academic Publishers Group, Dordrecht, 1988, doi: 10.1007/978-94-015-7793-9.  Google Scholar

[6]

J. Dynam. Differential Equations, 26 (2014), 1133-1169. doi: 10.1007/s10884-014-9403-0.  Google Scholar

[7]

SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211. 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]

Nonlinear Anal., 92 (2013), 82-95. doi: 10.1016/j.na.2013.06.017.  Google Scholar

[10]

J. Math. Anal. Appl., 411 (2014), 340-353. doi: 10.1016/j.jmaa.2013.08.064.  Google Scholar

[11]

J. Phys. A, 38 (2005), 8211-8223. doi: 10.1088/0305-4470/38/38/003.  Google Scholar

[12]

Commun. Pure Appl. Anal, 9 (2010), 583-610. 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]

Int. J. Bifurcat. Chaos, 25 (2015), 1550131. 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]

Math. Biosci., 88 (1988), 67-84. doi: 10.1016/0025-5564(88)90049-1.  Google Scholar

[17]

J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.  Google Scholar

[18]

J. Math. Chem., 51 (2013), 2001-2019. 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]

Nonlinearity, 27 (2014), 563-583. doi: 10.1088/0951-7715/27/3/563.  Google Scholar

[21]

Nonlinear Anal. Real World Appl., 14 (2013), 2002-2012. doi: 10.1016/j.nonrwa.2013.02.004.  Google Scholar

[22]

Nonlinearity, 21 (2008), 2121-2142. doi: 10.1088/0951-7715/21/9/013.  Google Scholar

[23]

Higher Education Press, Beijing; Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-22461-4.  Google Scholar

[24]

J. Theoret. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[25]

J. Math. Biol., 62 (2011), 291-331. doi: 10.1007/s00285-010-0332-1.  Google Scholar

[26]

Nonlinearity, 16 (2003), 1185-1201. 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]

Appl. Anal., 23 (1986), 63-76. doi: 10.1080/00036818608839631.  Google Scholar

[29]

American Mathematical Society, Providence, RI, 1992, Google Scholar

[1]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400

[2]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[3]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003

[4]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[5]

Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037

[6]

Tong Li, Nitesh Mathur. Riemann problem for a non-strictly hyperbolic system in chemotaxis. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021128

[7]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237

[8]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021071

[9]

Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021056

[10]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405

[11]

Zhisong Chen, Shong-Iee Ivan Su. Assembly system with omnichannel coordination. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021047

[12]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069

[13]

Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270

[14]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[15]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[16]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[17]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[18]

Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008

[19]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026

[20]

Francesca Bucci. Improved boundary regularity for a Stokes-Lamé system. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021018

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (4)

[Back to Top]