TY - JOUR
T1 - Poincaré Bifurcation from an Elliptic Hamiltonian of Degree Four with Two-Saddle Cycle
AU - Xiong , Yu’e
AU - Li , Wenyu
AU - Wang , Qinlong
JO - Journal of Nonlinear Modeling and Analysis
VL - 4
SP - 722
EP - 735
PY - 2023
DA - 2023/08
SN - 4
DO - http://doi.org/10.12150/jnma.2022.722
UR - https://global-sci.org/intro/article_detail/jnma/21908.html
KW - Perturbed Hamiltonian system, Poincaré bifurcation, Abelian integral, Chebyshev criterion.
AB - <p style="text-align: justify;">In this paper, we consider Poincaré bifurcation from an elliptic
Hamiltonian of degree four with two-saddle cycle. Based on the Chebyshev
criterion, not only one case in the Liénard equations of type (3, 2) is discussed
again in a different way from the previous ones, but also its two extended cases
are investigated, where the perturbations are given respectively by adding $εy(d_0 + d_2v^{2n} )\frac{∂}{∂y}$ with $n ∈ \mathbb{N}^ +$ and $εy(d_0 + d_4v^4 + d_2v^{2n+4})\frac{∂}{∂y}$ with $n = −1$ or $n ∈ \mathbb{N}^+,$ for small $ε &gt; 0.$ For the above cases, we obtain all the sharp upper
bound of the number of zeros for Abelian integrals, from which the existence
of limit cycles at most via the first-order Melnikov functions is determined.
Finally, one example of double limit cycles for the latter case is given.</p>