Cubic Liénard equations with quadratic damping having two antisaddles

Qualitative Theory of Dynamical Systems

Volumn 1, Issue 2, 2000, Pages 163-209

  Dumortier, Freddy ^a   Kooij, Robert E ^b   Li, Chengzhi ^c  

^a HASSELT UNIVERSITY   (Belgium)

^b KPN RESEARCH   (Netherlands)

^c PEKING UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0006661824     PISSN: 15755460     EISSN: 16623592     Source Type: Journal    
DOI: 10.1007/BF02969477     Document Type: Article

References (25)

1. A. A. Andronov et al., Theory of bifurcation of dynamical systems on the plane, translated by John Wiley and Sons, New York, 1973.
2. C. Christopher and N. Lloyd, Small-amplitude limit cycles in Liénard systems, Nonlinear differntial equations and applications, to appear.
3. F. Dumortier andC. Herssens, Polynomial Liénard equations near infinity, J. Diff. Equat. 153, 1999, 1-29.
4. F. Dumortier andC. Li, On the uniqueness of limit cycles surrounding one or more singularities in Liénard equations, Nonlinearity 9, 1996, 1489-1500.
5. F. Dumortier andC. Li, Quadratic Liénard equations with quadratic damping, J. Diff. Equat. 139, 1997, 41-59.
6. F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (1) Saddle Loop and Two Saddle Cycle, to appear in J. Diff. Equat.
7. F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (II) Cuspidal Loop, to appear in J. Diff. Equat.
8. F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (III) Global Center, Preprint.
9. F. Dumortier andC. Rousseau, Cubic Liénard equations with linear damping, Nonlinearity 3, 1990, 1015-1039.
10. R. E. Kooij and G. Giannakopoulos, Periodic orbits in planar systems modelling neural activity, to appear in Quarterly of Applied Mathematics.
11. A. Khibnik, B. Krauskopf andC. Rousseau, Global study of a family of cubic Lénard equations, Nonlinearity 11, 1998, 1505-1519.
12. R. E. Kooij andA. Zegeling, Qualitative properties of two-dimensional predatorprey systems, Nonlinear Analysis, TMA, Vol. 29, No. 6, 1997, 693-715.
13. R. E. Kooij andA. Zegeling, Limit cycles in quadratic systems with a weak focus and a strong focus, Kyungpook Math. Journal, Vol. 38, No. 2, 1998, 323-340.
14. A. Lins, W. de Melo and C. Pugh, On Liénard's equations, in "Geometry and Topology", Lect. Notes in Math. 597, 1977, 335-357.
15. S. Malo, Rigorous computer verification of planar vector field structure, Ph. D. thesis Cornell University, 1994.
16. G. S. Rychkov, Some criteria for the presence and absence of limit cycles in a second order dynamical system, Sibirsk. Mat. Z., 1966, 1425-1431. [in Russian].
17. 3(x))dy, of limit cycles encircling three critical points, Differents Uravn., Vol. 20 (1984), No. 11, 1906-1910. [in Russian].
18. 1(x, y)dx in the case of three critical points, Differents. Uravn. Vol. 21 (1985), No. 6, 991-997. [in Russian].
19. Wang Xian andR. E. Kooij, Limit cycles in cubic systems with a cusp, SIAM J. on Math. Anal., Vol. 23, No. 6, 1992, 1609-1622.
20. Zhang Pingguang andCai Suilin, Quadratic systems with a weak focus, Bull. Austral. Math. Soc., 44, 1991, 511-526.
21. Zhang Zhifen, Ding Tongren, Huang Wenzao andDong Zhenxi, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, Vol. 101, AMS, Providence, RI, 1992.
22. Zeng Xianwu, On the uniqueness of limit cycle of the Liénard equations, Sci. Sinica, A6 (1982), 584-592.
23. Zhang Zhifen, On the uniqueness of limit cycles of certain equations of nonlinear oscillations, Dokl. Akd. Nauk SSSR, 119, 1958, 659-662. [in Russian].
24. Zhang Zhifen, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations, Appl. Anal., 23, 1986, 63-76.
25. Zeng Xianwu, Zhang Zhifen andGao Suzhi, On the uniqueness of the limit cycle of the generalized Liénard equation, Bull. London Math. Soc., 26, 1994, 213-247.