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

Smale, 1991, Dynamics restrospective: great problem, attempts that failed, Physica D, 51, 261, 10.1016/0167-2789(91)90238-5 Arnold, 1990, Ten problems, Adv. Sov. Math, 1, 1 Ilyashenko, 1991 Arnold, 1977, Loss of stability of sell-oscillations close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11, 85, 10.1007/BF01081886 Li, 2003, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurcat. Chaos, 13, 47, 10.1142/S0218127403006352 Giné, 2007, On some open problems in planar differential systems and Hilbert’s 16th problem, Chaos, Solitons & Fractals, 31, 1118, 10.1016/j.chaos.2005.10.057 Li, 1989, Rotation symmetry groups of Hamiltonian system, Ann. Diff. Equs, 5, 25 Li, 1987, Bifurcations of limit cycles forming compound eyes in the cubic system, Chin. Ann. Math., 8B, 391 Li, 1991, Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system, Publ. Math., 35, 487, 10.5565/PUBLMAT_35291_13 Li, 2002, Investigations of bifurcations of limit cycles in a Z2-equivariant planar vector field of degree 5, Int. J. Bifurcat. Chaos, 12, 2137, 10.1142/S0218127402005698 Chan, 2001, Bifurcations of limit cycles in a Z3-equivariant planar vector field of degree 5, Int. J. Bifurcat. Chaos, 11, 2287, 10.1142/S0218127401003267 Li, 2005, On the control of parameters of distributions of limit cycles for a Z2-equivariant perturbed planar Hamiltonian polynomial vector field, Int. J. Bifurcat. Chaos, 15, 137, 10.1142/S0218127405012065 Li, 2002, Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5, Sci. China (Series A), 45, 817 Yu, 2005, Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields, Chaos, Solitons & Fractals, 24, 329, 10.1016/j.chaos.2004.09.036 Yu, 2006, Analysis on limit cycles of Zq-equivariant polynomial vector fields with degree 3 or 4, J. Math. Anal. Appl., 322, 51, 10.1016/j.jmaa.2005.08.068 Yao, 2007, Bifurcation of small limit cycles in Z5-equivariant planar vector fields of order 5, J. Math. Anal. Appl., 328, 400, 10.1016/j.jmaa.2006.05.056 Wu, 2007, On the limit cycles of a Hamiltonian under Z4-equivariant quintic perturbation, Chaos, Solitons & Fractals, 33, 298, 10.1016/j.chaos.2006.01.120 Chen, 2002, The number of limit cycles for a class quintic Hamiltonian systems under quintic perturbations, J. Austral. Math. Soc., 73, 37, 10.1017/S1446788700008466 Wang, 2005, Bifurcation of limit cycles in a quintic Hamiltonian system under sixth-order perturbation, Chaos, Solitons & Fractals, 26, 1317, 10.1016/j.chaos.2005.03.010 Zang, 2006, Bifurcations of limit cycles in a cubic system with cubic perturbations, Appl. Math. Comput., 176, 341, 10.1016/j.amc.2005.09.024 Carr, 1985, Abelian integrals and bifurcation theory, J. Differ. Equations, 59, 413, 10.1016/0022-0396(85)90148-2 Zhang, 1992 Guckenheimer, 1992 Perko, 1996 Pontryagin, 1934, On dynamical systems close to Hamiltonian systems, ZH. EKsp. Tero. Fiz, 4, 234 Ye, 1986 Melnikov, 1963, On the stability of the center for time periodic perturbations, Trans. Moscow. Math. Soc, 12, 1 Chen, 2004, The topological classification of plane diagram of a class quintic Hamiltonian system, Acta. Math. Sci, 24A, 737 Gudkov, 1974, The topology of real projective algebraic varieties, Russ. Math. Surv., 29, 1, 10.1070/RM1974v029n04ABEH001288 Abell, 1994