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Eight limit cycles around a center in quadratic Hamiltonian system with third-order perturbation. (English) Zbl 1270.34056

Summary: We show that generic planar quadratic Hamiltonian systems with third degree polynomial perturbation can have eight small-amplitude limit cycles around a center. We use higher-order focus value computation to prove this result, which is equivalent to the computation of higher-order Melnikov functions. Previous results have shown, based on first-order and higher-order Melnikov functions, that planar quadratic Hamiltonian systems with third degree polynomial perturbation can have five or seven small-amplitude limit cycles around a center. The result given in this paper is a further improvement.

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
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D10 Perturbations of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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