Perturbation from an asymmetric cubic Hamiltonian

Journal of Mathematical Analysis and Applications

Volumn 305, Issue 2, 2005, Pages 617-630

  Zhang, Tonghua ^a   Han, Maoan ^a   Zang, Hong ^a  

^a SHANGHAI JIAO TONG UNIVERSITY   (China)

Author keywords

Bifurcation; Hamiltonian system; Hilbert's 16th problem; Limit cycle; Stability

Indexed keywords

EID: 15844402332     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmaa.2004.12.044     Document Type: Article

References (36)

1. N.N. Bautin On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type Mat. Sb. (N.S.) 30 1952 181-196
2. N.N. Bautin Amer. Math. Soc. Transl. 100 1954
3. T. Bedford J. Swift New Directions in Dynamical Systems London Math. Soc. Lecture Note Ser. vol. 127 1988 Cambridge Univ. Press Cambridge
4. L.A. Cherkas V.A. Gaǐko Bifurcations of limit cycles of a quadratic system with two singular points and two parameters that rotate the field Differ. Uravn. 23 1987 1544-1553
5. C. Christopher J. Llibre A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals Proc. Amer. Math. Soc. 130 2002 2025-2030 (electronic)
6. F. Dumortier C. Li Perturbation from an elliptic Hamiltonian of degree four. IV. Figure eight-loop J. Differential Equations 188 2003 512-554
7. V.A. Gaiko Abelian integrals and limit cycles in polynomial dynamical systems Nonlinear Phenom. Complex Systems 6 2003 577-581
8. M. Galeotti M. Villarini Some properties of planar polynomial systems of even degree Ann. Mat. Pura Appl. 161 1992 299-313
9. A. Gasull J. Torregrosa A new approach to the computation of the Lyapunov constants. The geometry of differential equations and dynamical systems Comput. Appl. Math. 20 2001 149-177
10. L. Gavrilov I.D. Iliev Bifurcations of limit cycles from infinity in quadratic systems Canad. J. Math. 54 2002 1038-1064
11. M. Han Cyclicity of planar homoclinic loops and quadratic integrable systems Sci. China Ser. A 40 1997 1247-1258
12. M. Han J. Chen On the number of limit cycles in double homoclinic bifurcations Sci. China Ser. A 43 2000 914-928
13. M. Han S. Hu X. Liu On the stability of double homoclinic and heteroclinic cycles Nonlinear Anal. 53 2003 701-713
14. M. Han T. Zhang H. Zang On the number and distributions of limit cycles in a cubic system Internat. J. Bifur. Chaos 14 2004
15. M. Henrard F. Zanolin Bifurcation from a periodic orbit in perturbed planar Hamiltonian systems J. Math. Anal. Appl. 277 2003 79-103
16. I.D. Iliev L.M. Perko Higher order bifurcations of limit cycles J. Differential Equations 154 1999 339-363
17. J. Li Hilbert's 16th problem and bifurcations of planar polynomial vector fields Internat. J. Bifur. Chaos 13 2003 47-106
18. J. Li Q. Huang Bifurcations of limit cycles forming compound eyes in the cubic system Chinese Ann. Math. Ser. B 8 1989 391-403
19. 2-equivariant planar vector fields of degree 5 Internat. J. Bifur. Chaos 12 2002 2137-2157
20. J. Llibre Averaging theory and limit cycles for quadratic systems Rad. Mat. 11 2002/2003 215-228
21. J. Llibre D. Schlomiuk The geometry of quadratic differential systems with a weak focus of third order Canad. J. Math. 56 2004 310-343
22. N.G. Lloyd J.M. Pearson E. Saez I. Szanto A cubic Kolmogorov system with six limit cycles Comput. Math. 44 2002 445-455
23. V.G. Romanovski M. Robnik Computational of the center isochronism conditions for polynomial systems Differ. Uravn. 38 2002 195-200 (in Russian); translation in Differential Equations 38 (2002) 205-211
24. R. Roussarie Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem Prog. Math. vol. 164 1998 Birkhäuser Basel
25. Z. Huaiping C. Rousseau Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type J. Differential Equations 178 2002 325-436
26. D. Schlomiuk On the global analysis of the planar quadratic vector fields in: Proc. Second World Congr. of Nonlinear Analysts, Part 3 Athens, 1996 Nonlinear Anal. 30 1997 1429-1437
27. D.S. Shafer X. Wu A. Zegeling A numerical approach to bifurcations from quadratic centers J. Math. Anal. Appl. 202 1996 90-107
28. S. Smale Dynamics retrospective: Great problem, attempts that failed Phys. D 51 1991 261-273
29. Y. Ye Theory of Limit Cycles Transl. Math. Monogr. vol. 66 1986 Amer. Math. Soc. Boston
30. Yu. Ilyashenko Finiteness Theorem for Limit Cycles 1991 Amer. Math. Soc. Providence, RI
31. A. Zegeling R.E. Kooij The distribution of limit cycles in quadratic systems with four finite singularities J. Differential Equations 151 1999 373-385
32. T. Zhang H. Zang M. Han Bifurcations of limit cycles in a cubic system Chaos Solitons Fractals 20 2004 629-638
33. H. Zang T. Zhang W. Chen Bifurcations of limit cycles from quintic Hamiltonian system with a double figure eight loop J. Complexity 20 2004 544-560
34. H. Zang W. Chen T. Zhang Perturbation from a cubic Hamiltonian with three figure-eight loops Chaos Solitons Fractals 22 2004 61-74
35. T. Zhang H. Zang M. Han Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations Chaos Solitons Fractals 22 2004 1127-1138
36. H. Zoladek Eleven small limit cycles in a cubic vector field Nonlinearity 8 1995 843-860