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Cyclicity of homoclinic loops and degenerate cubic Hamiltonians. (English) Zbl 0959.34022
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.

MSC:
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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