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Higher order Melnikov function for a quartic hamiltonian with cuspidal loop

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  • 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 $g(x,y,\epsilon)$.

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

    Citation:

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