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Poincar{\'e} Bifurcation from an Elliptic Hamiltonian of Degree Four with Two-saddle Cycle
  
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KeyWord:Perturbed Hamiltonian system; Poincar{\'e} bifurcation; Abelian integral; Chebyshev criterion
Author NameAffiliation
Yu'e Xiong School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China 
Wenyu Li School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China 
Qinlong Wang School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China
Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin, Guangxi 541004, China 
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Abstract:
      In this paper, we consider Poincar{\'e} bifurcation from an elliptic Hamiltonian of degree four with two-saddle cycle. Based on the Chebyshev criterion, not only one case in the Li{\'e}nard 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 $\varepsilon y(d_0 + d_2 v^{2n})\frac{\partial }{{\partial y}}$ with $ n\in \mathbb{N^+}$ and $\varepsilon y(d_0 + d_4 {v^4}+ d_2 v^{2n+4})\frac{\partial }{{\partial y}}$ with $n=-1$ or $ n\in \mathbb{N^+}$, for small $\varepsilon > 0$. 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.