Poincaré Bifurcation from an Elliptic Hamiltonian of Degree Four with Two-Saddle Cycle (Journal of Nonlinear Modeling and Analysis, 2024)
In this paper, we consider Poincaré bifurcation from an ellipticHamiltonian of degree four with two-saddle cycle. Based on the Chebyshevcriterion, not only one case in the Liénard equations of type (3, 2) is discussedagain in a different way from the previous ones, but also its two extended casesare 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 $ε > 0.$ For the above cases, we obtain all the sharp upperbound of the number of zeros for Abelian integrals, from which the existenceof 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.