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A Simple Proof for the Unicity of the Limit Cycle in the Bogdanov-Takens System

Published online by Cambridge University Press:  20 November 2018

Chengzhi Li ,
Christiane Rousseau  and
Xian Wang
Chengzhi Li
Affiliation:
Department of mathematics, peking university, beijing, PRC
Christiane Rousseau
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128 Succ. A, Montréal Qué., H3C 3J7, Canada
Xian Wang
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, PRC
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Abstract

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We show that the Bogdanov-Takens system has at most one limit cycle. Similarly we show that the maximum number of limit cycles in the universal unfolding of the symmetric cusp of order 2 (resp. 3) is one (resp. 2). The proof uses the elementary technique of Liénard's equation, yielding a global result for all values of the parameters.

Keywords

34C0534C3539
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 1 , 01 March 1990 , pp. 84 - 92
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Bogdanov, R. I., Bifurcation of a limit cycle for a family of vector fields in the plane, Trudy Seminar Petrovskii, (1976), (in Russian), Sel. Math. Sov., 1 (1981), 373–387 (in English).Google Scholar
2. Bogdanov, R. I., Versai deformation of a singular point of a vector field on the plane in the case of zero eigenvalues, Trudy Seminar Petrovskii, (1976), (in Russian), Sel. Math. Sov., 1 (1981), 388–421 (in English).Google Scholar
3. Carr, J., Applications of Centre Manifold Theory, Springer-Verlag New York, Heidelberg, Berlin, (1981).Google Scholar
4. Cherkas, L. A., Estimation of the number of limit cycles of autonomous systems, Differential Equations, 13 (1977), 529–547.Google Scholar
5. Cushman, R. and Sanders, J., Abelian integrals and global Hopf bifurcations, Springer Lecture Notes in Mathematics 1125 (1985), 87–98.Google Scholar
6. Dumortier, F., R. Roussarie and Sotomayor, J., Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynamical Systems, 7 (1987), 375–413.Google Scholar
7. Horozov, E. I., Versai deformations of equivariant vector fields under symmetries of order2 and 3, Trudy Seminar Petrovskii, 5 (1979), 163–192.Google Scholar
8. Li, Chengzhi and Rousseau, C., Codimension 2 symmetric homoclinic bifurcations and application, preprint, to appear in Can. J. of Math.Google Scholar
9. Mardesic, P., The number of limit cycles of polynomial deformations of a Hamiltonian vector field, preprint 1988.Google Scholar
10. Perko, L. M., Rotated vector fields and the global behaviour of limit cycles for a class of quadratic systems in the plane, J. of Differential Equations, 18 (1975), 63–86.Google Scholar
11. Perko, L. M., Global analysis of Bogdanov's system, preprint, 1988.Google Scholar
12. Petrov, G. S., Elliptic integrals and their nonoscillation, Funct. Anal. Appl., 20 1986, 37–40.Google Scholar
13. Roussarie, R., Déformations génériques des cusps, (preprint, 1986).Google Scholar
14. Rychkov, G. S., The maximal number of limit cycles of the system Is equal to two, Differential Equations, 11 (1975), 301–302.Google Scholar
15. Takens, F., Forced oscillations and bifurcations, in Applications of global analysis I, Comm. Math. Inst. Rijksuniversiteit Utrecht, (1974), 1-59.Google Scholar
16. Ye, Yanqian, Theory of limit cycles, Translations of Mathematical Monographs, AMS, (1986).Google Scholar
17. Zhang, Zhifen, On the existence of exactly two limit cycles for the Liénard equation, Acta Math. Sinica, 24 (1981), 710–716 (in Chinese) (see [16], theorem 7.2).Google Scholar
18. Zhang, Zhifen, On the uniqueness of limit cycles for some equations of non-linear oscillations, Dokl. Akad. Nauk USSR, 119 (1958), 659–662.Google Scholar
19. Zhang, Zhifen, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations, Applicable Analysis, 23 (1986), 63–76.Google Scholar