American Institute of Mathematical Sciences

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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
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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

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