Cyclicity of homoclinic loops and degenerate cubic Hamiltonians
New conditions for a planar system of the form \[ \dot x=H_y+\varepsilon f(x,y,\varepsilon ,\delta),\quad \dot y=-H_x+\varepsilon g(x,y,\varepsilon ,\delta), \] with \(H,f,g\in C^{\infty}\), \(\varepsilon\geq 0\) is small, \(\delta\in\mathbb R^n\) with \(n>3\) to have for \(\varepsilon +|\delta -\delta _0|\) small at most two limit cycles near a homoclinic loop of the unperturbed system (\(\varepsilon =0\)) are obtained. As an application it is proved that a homoclinic loop of a degenerate cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations.
perturbations of planar Hamiltonian systems
scientific article; zbMATH DE number 1531239
scientific article