Abstract
The weakened Hilberts 16th problem for symmetric planar perturbed polynomial Hamiltonian systems is considered. An example of Z8-equivariant planar perturbed Hamiltonian systems is constructed. By using bifurcation theory of planar dynamical systems and the method of detection functions, under the help of numerical analysis, it is shown that there exist parameter groups such that this perturbed Hamiltonian polynomial vector field of degree 7 has at least72=49 limit cycles with Z8 symmetry. It also gives rise to different configurations of limit cycles forming compound eyes.
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References
M. L. Abell J. P. Braselton (1994) Maple V: by Example AP Professional Boston
V. I. Arnold (1983) Geometric Methods in Theory of Ordinary Differential Equations Springer-Verlag New York
H. Giacomini J. Llibre M. Viano (1996) ArticleTitleOn the nonexistence, existence and uniqueness of limit cycles Nonlinearity 9 501–516
J. Guckenheimer P. Holmes (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of vector Fields Springer-Verlag New York
D. A. Gudkov (1974) ArticleTitleThe topology of real projective algebraic varieties Russian Math. Surveys 29 4 1–79
D. Hilbert (1976) ArticleTitleMathematical problems In Proceeding of Symposia in Pure Mathematics 28 1–34
Li. Jibin Li. Cunfu (1985) ArticleTitlePlanar cubic Hamiltonian systems and distribution of limit cycles of E3 Acta. Math. Sinica 28 4 509–521
Li. Jibin Huang Qimin (1987) ArticleTitleBifurcations of limit cycles forming compound eyes in the cubic system Chinese Ann. of Math. 8B 391–403
Jibin, Li., and Zhengrong, Liu. (1992). Bifurcation set and compound eyes in a perturbed cubic Hamiltonian system. In Ordinary and Delay Differential π,Pitman Research Notes in Math. Series 272, Longman, England, pp. 116–128.
Li. Jibin Zhao Xiaohua. (1989) ArticleTitleRotation symmetry groups of planar Hamiltonian systems Ann.of Diff. Eqs. 5 25–33
Li. Jibin Liu. Zhenrong (1991) Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system Publications Mathmatiques Spain 487–506
J. Llibre Zhang Xiang (2001) ArticleTitleOn the number of limit cycles for some perturbed Hamiltonian polynomial systems J. Dyn. Conti. Disc. Imp. Sys. Ser. A 8 161–182
N.G. Lloyd (1988) Limit cycles of polynomial systems T. Bedford J. Swift (Eds) New Directions in Dynamical Systems London Mathematical Society Lecture Notes London 192–234
S. Smale (1998) ArticleTitleMathematical problems for the next century The Math Intelligencer 20 2 7–15
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Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.
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Li, J., Zhang, M. Bifurcations of Limit Cycles in a Z8-Equivariant Planar Vector Field of Degree 7. J Dyn Diff Equat 16, 1123–1139 (2004). https://doi.org/10.1007/s10884-004-7835-7
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DOI: https://doi.org/10.1007/s10884-004-7835-7