November  2002, 8(4): 995-1018. doi: 10.3934/dcds.2002.8.995

Higher order Melnikov function for a quartic hamiltonian with cuspidal loop

1. 

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

2. 

Department of Mathematics, Zhongshan (Sun Yat-sen) University, 510275, Guangzhou, China

Received  February 2001 Revised  April 2002 Published  July 2002

Consider the polynomial perturbations of Hamiltonian vector field

$X_\epsilon=(H_y+\epsilon f(x,y,\epsilon))\frac{\partial}{\partial x}+ (-H_x+\epsilon g(x,y,\epsilon))\frac{\partial}{\partial y},$

where the Hamiltonian $H(x,y)=\frac{1}{2}y^2+U(x)$ has one center and one cuspidal loop, $deg U(x)=4$. In present paper we find an upper bound for the number of zeros of the $k$th order Melnikov function $M_k(h)$ for arbitrary polynomials $f(x,y,\epsilon)$ and $g(x,y,\epsilon)$.

Citation: Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995
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