Limit cycles for the kukles system

Journal of Dynamical and Control Systems

Volumn 14, Issue 2, 2008, Pages 283-298

  Zang, Hong ^a   Zhang, Tonghua ^a   Tian, Yu Chu ^b   Tadé, Moses O ^a  

^a CURTIN UNIVERSITY   (Australia)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Double homoclinic bifurcation; Focus quantity; Homoclinic bifurcation; Kukles system; Limit cycles; Stability

Indexed keywords

DOUBLE HOMOCLINIC BIFURCATION; HOMOCLINIC BIFURCATION; KUKLES SYSTEM; LIMIT CYCLES; QUALITATIVE ANALYSIS;

BIFURCATION (MATHEMATICS); CONTROL SYSTEM STABILITY; PERTURBATION TECHNIQUES;

CONTROL SYSTEMS;

EID: 42149088820     PISSN: 10792724     EISSN: 15738698     Source Type: Journal    
DOI: 10.1007/s10883-008-9036-x     Document Type: Article

References (20)

1. C. J. Christopher and N. G. Lloyd, On the paper of Jin and Wang concering the conditions for a center in certain cubic system. Bull. London Math. Soc. 22 (1990), No. 9, 5-12.
2. C. Christopher and J. Devlin, Isochronous centers in planar polynomial systems. SIAM J. Math. Anal. 28 (1997), 162-177.
3. M. Han and J. Chen, On the number of limit cycles in double homoclinic bifurcations. Sci. China, Ser. A 43 (2000), No. 9, 914-928.
4. M. Han, S. Hu, and X. Liu, On the stability of double homoclinic and heteroclinic cycles. Nonlinear Anal. 53 (2003), 701-713.
5. M. Han, T. Zhang, and H. Zang, On the number and distribution of limit cycles in a cubic system. Int. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), No. 12, 4285-4292.
6. M. Han and T. Zhang, Some bifurcation methods of finding limit cycles. Math. Biosci. Eng. 3 (2006), No. 1, 67-77.
7. X. Jin and D. Wang, On the conditions of Kukles for the existence of a center. Bull. London Math. Soc. 22 (1990), No. 9, 1-4.
8. I. S. Kukles, Necessary and sufficient conditions for the existence of a center. Dokl. Akad. Nauk SSSR 42 (1944), 160-163 (in Russian).
9. Y. Liu and W. Huang, A new method to determine isochronous center conditions for polynomial differential systems. Bull. Sci. Math. 127 (2003), 133-148.
10. R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Bras. Mat. 17 (1986), 67-101.
11. C. Rousseau, D. Schlomiuk, and P. Thibaudeau, The centers in the reduced Kukles system. Nonlinearity 8 (1995), 541-569.
12. C. Rousseau and B. Toni, Local bifurcations of critical periods in the reduced kukles system. Can. J. Math. 49 (1997), No. 2, 338-358.
13. A. P. Sadovskii, Solution of the center-focus problem for cubic systems in nonlinear oscillations. Differ. Uravnen. 33 (1997), 236-244 (in Russian).
14. A. P. Sadovskii, Cubic systems of nonlinear oscillations with seven limit cycles. Differ. Uravnen. 39 (2003), 472-481 (in Russian).
15. V. A. Gaiko and W. T. van Horssen, Global bifurcation of limit and separatrix cycles in a generalized Lienard system. Nonlinear Anal. 59 (2004), 189-198.
16. Y. Wu, G. Chen, and X. Yang, Kukles system with two fine foci. Ann. Differ. Equations 15 (1999), No. 4, 422-437.
17. W. Ye and Y. Ye, Qualitative theory of the Kukles system, I. Number of critical points. Ann. Differ. Equations 17 (2001), 275-286.
18. Y. Ye, Theory of limit cycles. Trans. Math. Monogr. 66 (1986).
19. H. Zang, T. Zhang, and C. Wencheng, Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop. J. Complexity 20 (2004), No. 4, 544-560.
20. T. Zhang, M. Han, and H. Zang, Perturbation from an asymmetric cubic Hamiltonian. J. Math. Anal. Appl. 305 (2005), 617-630.