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- Title
Estimating the Number of Zeros of Abelian Integrals for the Perturbed Cubic Z4-Equivariant Planar Hamiltonian System.
- Authors
Chen, Aiyong; Huang, Wentao; Xia, Yonghui; Zhang, Huiyang
- Abstract
We analyze the dynamics of a class of Z 4 -equivariant Hamiltonian systems of the form ż = (a + b | z | 2) z i + z ¯ 3 i , where z is complex, the time t is real, while a and b are real parameters. The topological phase portraits with at least one center are given. The finite generators of Abelian integral I (h) = ∮ Γ h g (x , y) d x − f (x , y) d y are obtained, where Γ h is a family of closed ovals defined by H (x , y) = − a 2 y 2 − a 2 x 2 − 1 + b 4 x 4 − 1 + b 4 y 4 − b − 3 2 x 2 y 2 = h , h ∈ Σ , Σ is the open interval on which Γ h is defined, f (x , y) , g (x , y) are real polynomials in x and y with degree n. We give an estimation of the number of isolated zeros of the corresponding Abelian integral by using its algebraic structure. We show that for the given polynomials f (x , y) and g (x , y) in x and y with degree n , the number of the limit cycles of the perturbed Z 4 -equivariant Hamiltonian system does not exceed 8 3 n − 3 (taking into account the multiplicity).
- Subjects
ABELIAN functions; HAMILTONIAN systems; LIMIT cycles; POLYNOMIALS
- Publication
International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, 2023, Vol 33, Issue 7, p1
- ISSN
0218-1274
- Publication type
Article
- DOI
10.1142/S0218127423500852