Detection function method and its application to a class of quintic Hamiltonian systems with quintic perturbations

Applied Mathematics and Computation

Volumn 191, Issue 2, 2007, Pages 490-503

  Zhou, Hongxian ^a   Xu, Wei ^a   Zhao, Xiaoshan ^a   Li, Shuang ^a  

^a NORTHWESTERN POLYTECHNICAL UNIVERSITY   (China)

Author keywords

Detection function; Hopf bifurcation; Limit cycle; Quintic Hamiltonian system

Indexed keywords

DYNAMICAL SYSTEMS; FUNCTION EVALUATION; HOPF BIFURCATION; PARAMETER ESTIMATION; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

DETECTION FUNCTION; LIMIT CYCLES; QUINTIC HAMILTONIAN SYSTEMS;

HAMILTONIANS;

EID: 34548118258     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2007.02.151     Document Type: Article

References (30)

1. Smale S. Dynamics restrospective: great problem, attempts that failed. Physica D 51 (1991) 261-273
2. Arnold V.I. Ten problems. Adv. Sov. Math 1 (1990) 1-8
3. Ilyashenko Yu. Finiteness Theorem for Limit Cycles (1991), American Mathematical Society, Providence, RI
4. Arnold V.I. Loss of stability of sell-oscillations close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11 (1977) 85-92
5. Li J. Hilbert's 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurcat. Chaos 13 1 (2003) 47-106
6. Giné J. On some open problems in planar differential systems and Hilbert's 16th problem. Chaos, Solitons & Fractals 31 5 (2007) 1118-1134
7. Li J., and Zhao X. Rotation symmetry groups of Hamiltonian system. Ann. Diff. Equs 5 (1989) 25-33
8. Li J., and Huang Q. Bifurcations of limit cycles forming compound eyes in the cubic system. Chin. Ann. Math. 8B (1987) 391-403
9. Li J., and Liu Z. Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system. Publ. Math. 35 (1991) 487-506 (in Spanish)
10. 2-equivariant planar vector field of degree 5. Int. J. Bifurcat. Chaos 12 10 (2002) 2137-2157
11. 3-equivariant planar vector field of degree 5. Int. J. Bifurcat. Chaos 11 8 (2001) 2287-2298
12. 2-equivariant perturbed planar Hamiltonian polynomial vector field. Int. J. Bifurcat. Chaos 15 1 (2005) 137-155
13. 6-equivariant planar vector field of degree 5. Sci. China (Series A) 45 7 (2002) 817-826
14. 2-equivariant vector fields. Chaos, Solitons & Fractals 24 1 (2005) 329-348
15. q-equivariant polynomial vector fields with degree 3 or 4. J. Math. Anal. Appl. 322 (2006) 51-65
16. 5-equivariant planar vector fields of order 5. J. Math. Anal. Appl. 328 (2007) 400-413
17. 4-equivariant quintic perturbation. Chaos, Solitons & Fractals 33 1 (2007) 298-307
18. Chen G., Wu Y., and Yang X. The number of limit cycles for a class quintic Hamiltonian systems under quintic perturbations. J. Austral. Math. Soc. 73 (2002) 37-53
19. Wang S., and Yu P. Bifurcation of limit cycles in a quintic Hamiltonian system under sixth-order perturbation. Chaos, Solitons & Fractals 26 5 (2005) 1317-1335
20. Zang H., Zhang T., and Han M. Bifurcations of limit cycles in a cubic system with cubic perturbations. Appl. Math. Comput. 176 (2006) 341-358
21. Carr J., Chow S.N., and Hale J.K. Abelian integrals and bifurcation theory. J. Differ. Equations 59 (1985) 413-417
22. Zhang Z., Ding T., Huang W., and Dong Z. Qualitative theory of differential equations [Transl Math Monographs 101] (1992), American Mathematical Society, Providence, RI
23. Guckenheimer J., and Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. second ed. (1992), Springer-Verlag, New York
24. Perko L.M. Differential Equations and Dynamical Systems (1996), Springer-Verlag, New York
25. Pontryagin L.S. On dynamical systems close to Hamiltonian systems. ZH. EKsp. Tero. Fiz 4 8 (1934) 234-238 (in Russian)
26. Ye Y. Theory of Limit Cycles [Transl Math Monographs 66] (1986), American Mathematical Society, Providence, RI
27. Melnikov V.K. On the stability of the center for time periodic perturbations. Trans. Moscow. Math. Soc 12 (1963) 1-57
28. Chen G., and Yang X. The topological classification of plane diagram of a class quintic Hamiltonian system. Acta. Math. Sci 24A 6 (2004) 737-751 (in Chinese)
29. Gudkov D.A. The topology of real projective algebraic varieties. Russ. Math. Surv. 29 4 (1974) 1-34
30. Abell M.L., and Braselton J.P. Maple V: By Example (1994), AP Professional, Boston