Bifurcations of limit cycles in a Z 6 -equivariant planar vector field of degree 5
TLDR
In this paper, a concrete numerical example of Z 6 -equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions.Abstract:
A concrete numerical example of Z 6 -equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H (2k + 1)≥(2 k + 1) 2 -1 for the perturbed Hamiltonian systems.read more
Citations
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Journal ArticleDOI
Hilbert's 16th problem and bifurcations of planar polynomial vector fields
TL;DR: The progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 15th problem and bifurcations of planar vector fields is discussed.
Journal ArticleDOI
Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields
TL;DR: In this article, the existence of 12 small limit cycles for cubic order Z2-equivariant vector fields with saddle points and symmetric focus points has been proved, which is a new result in the second part of the 16th Hilbert problem.
Journal ArticleDOI
Lower bounds for the Hilbert number of polynomial systems
Maoan Han,Jibin Li +1 more
TL;DR: In this paper, it was shown that H(m) grows at least as rapidly as 12ln2(m+2)2ln(m + 2) for all large m. The results obtained in this paper improve all existing results for all m⩾7 based on some known results for m=3,4,5,6.
Journal ArticleDOI
New Results on the Study of Zq-Equivariant Planar Polynomial Vector Fields
Jibin Li,Jibin Li,Yirong Liu +2 more
TL;DR: In this article, it was shown that a 2-equivariant planar polynomial vector field with two elementary foci has at most 12 limit cycles with the scheme of the first six Lyapunov constants and that the Hilbert number H(3) ≥ 13.
Journal ArticleDOI
Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations
TL;DR: In this paper, the number of limit cycles of a cubic system with quartic perturbations is investigated and their distributions are studied by using the methods of bifurcation theory and qualitative analysis, which gives rise to the conclusion that H(4)⩾15, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem.
References
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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
TL;DR: In this article, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
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TL;DR: In this paper, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
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