Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system

Journal of Differential Equations

Volumn 246, Issue 1, 2009, Pages 129-163

  Han, Maoan ^a   Zang, Hong ^a   Yang, Junmin ^a  

^a SHANGHAI NORMAL UNIVERSITY   (China)

Author keywords

Bifurcation; Cuspidal loop; Hamiltonian system; Limit cycle; Melnikov function

Indexed keywords

EID: 55049105396     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.06.039     Document Type: Article

References (15)

1. Dumortier F., and Li C. Perturbations from an elliptic Hamiltonian of degree four, II. Cuspidal loop. J. Differential Equations 175 2 (2001) 209-243
2. Han M. Bifurcation theory of limit cycles of planar systems. In: Canada A., Drabek P., and Fonda A. (Eds). Handbook of Differential Equations, vol. 3, Ordinary Differential Equations (2006), Elsevier
3. Han M. Bifurcations of invariant tori and subharmonic solutions for periodic perturbed systems. Sci. China Ser. A 37 11 (1994) 1325-1336
4. Han M. Liapunov constants and Hopf cyclicity of Lienard systems. Ann. Differential Equations 15 2 (1999) 113-126
5. Han M. Cyclicity of planar homoclinic loops and quadratic integrable systems. Sci. China Ser. A 40 12 (1997) 1247-1258
6. Han M., and Chen J. On the number of limit cycles in double homoclinic bifurcations. Sci. China Ser. A 43 9 (2000) 914-928
7. M. Han, J. Yang, A. Tarta, Y. Gao, Limit cycles near homoclinic and heteroclinic loops, J. Dynam. Differential Equations (2008), doi:10.1007/s10884-008-9108-3
8. Han M., Ye Y., and Zhu D. Cyclicity of homoclinic loops and degenerate cubic Hamiltonians. Sci. China Ser. A 42 6 (1999) 605-617
9. Han M., and Zhang T. Some bifurcation methods of finding limit cycles. Math. Biosci. Eng. 3 1 (2006) 67-77
10. Jiang Q., and Han M. Melnikov functions and perturbation of planar Hamiltonian system. Chin. Ann. Math. Ser. B 20 2 (1999) 233-246
11. Li J. Hilbert's 16th problem and bifurcations of planar polynomial vector fields. Internat. J. Bifur. Chaos 13 1 (2003) 47-106
12. Li W., Llibre J., and Zhang X. Melnikov functions for period annulus, nondegenerate centers, heteroclinic and homoclinic cycles. Pacific J. Math. 213 1 (2004) 49-77
13. Roussarie R. On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Brasil. Mat. 17 2 (1986) 67-101
14. Schlomiuk D. Algebraic and geometric aspects of the theory of polynomial vector fields. In: Schlomiuk D. (Ed). Bifurcations and Periodic Orbits of Vector Fields. NATO ASI Ser. C vol. 408 (1993), Kluwer Academic, London 429-467
15. Zhao Y., and Zhang Z. Abelian integrals for cubic vector fields. Ann. Mat. Pura Appl. (4) 176 (1999) 251-272