Higher order Melnikov function for a quartic hamiltonian with cuspidal loop

Yulin Zhao; Siming Zhu

doi:10.3934/dcds.2002.8.995

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2002, Volume 8, Issue 4: 995-1018. Doi: 10.3934/dcds.2002.8.995

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Higher order Melnikov function for a quartic hamiltonian with cuspidal loop

Yulin Zhao^1, and
Siming Zhu^2,

1.
Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275
2.
Department of Mathematics, Zhongshan (Sun Yat-sen) University, 510275, Guangzhou

Received: February 2001

Revised: April 2002

Published: October 2002

Abstract / Introduction Related Papers Cited by

Abstract

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 $g(x,y,\epsilon)$.

Keywords:

Abelian integrals,
k-th order Melnikov function.

Mathematics Subject Classification: 34C07, 34C08, 37G15.

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