Research ArticleOpen AccessGoogle Scholar indexed
On Canard Homoclinic of a Liénard Perturbation System
- 1
Applied Mathematics·Volume 02 (2011)·Pages 1221–1224·Published 11 October 2011·DOI10.4236/am.2011.210170
Copy link · social · email
Abstract
The classification on the orbits of some Liénard perturbation system with several parameters, which is relation to the example in [1] or [2], is discussed. The conditions for the parameters in order that the system has a unique limit cycle, homoclinic orbits, canards or the unique equilibrium point is globally asymptotic stable are given. The methods in our previous papers are used for the proofs.
KeywordsLiénard SystemCanardsLimit CyclesHomoclinic OrbitsGlobal Asymptotic Stability
- D. Changming, “The Homoclinic Orbits in the Liénard Plane,” Journal of Mathematical Analysis and Applica-tions, Vol. 191, 1995, pp. 26-39.
- B. Rachid, “Ca-nards Homocliniques,” the 7-th International Colloqium on Differential Equations in Bulgaria, Plovdiv, 1997.
- M. Hayashi, “A Geometric Condition for the Existence of the Homoclinic Orbits of Liénard Systems,” International Journal of Pure and Applied Mathematics, Vol. 66, No.1, 2011, pp. 53-60.
- E. Beno?t, “Systèmes Lents-Rapides Dans et Leurs a Canard Orbits,” Asterisque, Vol. 109-110, 1983, pp. 159-191.
- Z. Zhang, et al., “Qualitative Theory of Differential Equations,” Translate Mathematical Monographs in AMS, Vol. 102, 1992, p. 236.
- J. Graef, “On the Generalized Liénard Equation with Negative Damping,” Journal of Differential Equations, Vol. 12, 1992, pp. 34-62. doi:10.1016/0022-0396(72)90004-6
- M. Hayashi, “On Uniqueness of the Closed Orbit of the Liénard Sys-tem,” Mathematica Japonicae, Vol. 46, No. 3, 1997, pp. 371-376.
- A. Gasull and A. Guillamon, “Non-Existence of Limit Cycles for Some Predator-Prey Systems,” Proceedings of Equadiff’91, World Scientific, Singapore, 1993, pp. 538-546.
- M. Hayashi, “Non-Existence of Homoclinic Orbits and Global Asymptotic Stability of FitzHugh-Nagumo System,” Vietnam Journal of Mathematics, Vol. 27, No. 4, 1999, pp. 335-343.
- J. Sugie, “Non-Existence of Periodic Solutions for the FitzHugh Nerve System,” Quarterly Journal of Applied Mathematics, Vol. 49, 1991, pp. 543-554.