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

[2]

J. C. Artés, J. Llibre, J. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comput. Simulation, 95 (2014), 13-22. doi: 10.1016/j.matcom.2013.02.007.

[3]

V. Carmona, S. Fernández-García, E. Freire and F. Torres, Melnikov theory for a class of planar hybrid systems, Phys. D, 248 (2013), 44-54. doi: 10.1016/j.physd.2013.01.002.

[4]

Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems, Nonlinear Anal., 69 (2008), 3610-3628. doi: 10.1016/j.na.2007.09.045.

[5]

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

[6]

N. Forcadel, A. Ghorbel and S. Walha, Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model, J. Dynam. Differential Equations, 26 (2014), 1133-1169. doi: 10.1007/s10884-014-9403-0.

[7]

E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211. doi: 10.1137/11083928X.

[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.

[9]

S. Huan and X. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics, Nonlinear Anal., 92 (2013), 82-95. doi: 10.1016/j.na.2013.06.017.

[10]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types, J. Math. Anal. Appl., 411 (2014), 340-353. doi: 10.1016/j.jmaa.2013.08.064.

[11]

T. W. Hwang and H. J. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions, J. Phys. A, 38 (2005), 8211-8223. doi: 10.1088/0305-4470/38/38/003.

[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, Commun. Pure Appl. Anal, 9 (2010), 583-610. doi: 10.3934/cpaa.2010.9.583.

[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. 

[14]

F. Jiang, J. Shi and J. Sun, On the number of limit cycles for discontinuous generalized linéard polynomial differential systems, Int. J. Bifurcat. Chaos, 25 (2015), 1550131. doi: 10.1142/S021812741550131X.

[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.

[16]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci., 88 (1988), 67-84. doi: 10.1016/0025-5564(88)90049-1.

[17]

C. Z. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.

[18]

P. Liu, J. P. Shi, Y. W. Wang and X. H. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model, J. Math. Chem., 51 (2013), 2001-2019. doi: 10.1007/s10910-013-0196-x.

[19]

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

[20]

J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583. doi: 10.1088/0951-7715/27/3/563.

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

[22]

J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities, Nonlinearity, 21 (2008), 2121-2142. doi: 10.1088/0951-7715/21/9/013.

[23]

A. C. J. Luo, Discontinuous Dynamical Systems, Higher Education Press, Beijing; Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-22461-4.

[24]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.

[25]

J. F. Wang, J. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331. doi: 10.1007/s00285-010-0332-1.

[26]

D. M. Xiao and Z. F. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey systems, Nonlinearity, 16 (2003), 1185-1201. doi: 10.1088/0951-7715/16/3/321.

[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, (). 

[28]

Z. F. Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations, Appl. Anal., 23 (1986), 63-76. doi: 10.1080/00036818608839631.

[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, Providence, RI, 1992,

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

[2]

J. C. Artés, J. Llibre, J. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comput. Simulation, 95 (2014), 13-22. doi: 10.1016/j.matcom.2013.02.007.

[3]

V. Carmona, S. Fernández-García, E. Freire and F. Torres, Melnikov theory for a class of planar hybrid systems, Phys. D, 248 (2013), 44-54. doi: 10.1016/j.physd.2013.01.002.

[4]

Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems, Nonlinear Anal., 69 (2008), 3610-3628. doi: 10.1016/j.na.2007.09.045.

[5]

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

[6]

N. Forcadel, A. Ghorbel and S. Walha, Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model, J. Dynam. Differential Equations, 26 (2014), 1133-1169. doi: 10.1007/s10884-014-9403-0.

[7]

E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211. doi: 10.1137/11083928X.

[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.

[9]

S. Huan and X. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics, Nonlinear Anal., 92 (2013), 82-95. doi: 10.1016/j.na.2013.06.017.

[10]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types, J. Math. Anal. Appl., 411 (2014), 340-353. doi: 10.1016/j.jmaa.2013.08.064.

[11]

T. W. Hwang and H. J. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions, J. Phys. A, 38 (2005), 8211-8223. doi: 10.1088/0305-4470/38/38/003.

[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, Commun. Pure Appl. Anal, 9 (2010), 583-610. doi: 10.3934/cpaa.2010.9.583.

[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. 

[14]

F. Jiang, J. Shi and J. Sun, On the number of limit cycles for discontinuous generalized linéard polynomial differential systems, Int. J. Bifurcat. Chaos, 25 (2015), 1550131. doi: 10.1142/S021812741550131X.

[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.

[16]

Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Math. Biosci., 88 (1988), 67-84. doi: 10.1016/0025-5564(88)90049-1.

[17]

C. Z. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations, 252 (2012), 3142-3162. doi: 10.1016/j.jde.2011.11.002.

[18]

P. Liu, J. P. Shi, Y. W. Wang and X. H. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model, J. Math. Chem., 51 (2013), 2001-2019. doi: 10.1007/s10910-013-0196-x.

[19]

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

[20]

J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583. doi: 10.1088/0951-7715/27/3/563.

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

[22]

J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities, Nonlinearity, 21 (2008), 2121-2142. doi: 10.1088/0951-7715/21/9/013.

[23]

A. C. J. Luo, Discontinuous Dynamical Systems, Higher Education Press, Beijing; Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-22461-4.

[24]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0.

[25]

J. F. Wang, J. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331. doi: 10.1007/s00285-010-0332-1.

[26]

D. M. Xiao and Z. F. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey systems, Nonlinearity, 16 (2003), 1185-1201. doi: 10.1088/0951-7715/16/3/321.

[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, (). 

[28]

Z. F. Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations, Appl. Anal., 23 (1986), 63-76. doi: 10.1080/00036818608839631.

[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, Providence, RI, 1992,

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