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Perturbations from a kind of quartic Hamiltonians under general cubic polynomials. (English) Zbl 1192.34044

The authors extend results by Dumortier and Li on perturbations of a class of quartic Hamiltonian planar vector fields \(X\) (or Li√©nard equations), admitting now arbitrary cubic perturbations. They prove that if \(X\) has more than one equilibrium, the number of limit cycles of the perturbed system which surround only one center of \(X\) does not exceed twelve. If \(X\) has a saddle loop, they show the existence of a perturbation exhibiting at least three limit cycles, and, moreover, they present a subclass of \(X\) whose perturbations produce at most three limit cycles near the original homoclinic loop. The proofs use standard methods involving Abelian integrals, Picard-Fuchs equations, ratios of (derivatives of) abelian integrals and corresponding Riccati equations. For the expansion of the Abelian integrals near the homoclinic loop, the coefficients up to order four are determined by tedious calculations. As for the zeros of Abelian integrals, the authors apply theorems on the number of intersection points of planar curves that are similar to Khovanskii’s results.

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
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
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References:

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