Poincaré Bifurcation from an Elliptic Hamiltonian of Degree Four with Two-Saddle Cycle

Abstract

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 $ε > 0.$ 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.