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The cyclicity of period annuli of a class of quartic Hamiltonian systems. (Chinese. English summary) Zbl 1389.37038

Summary: In this paper, we prove that the cyclicity of period annuli for system \[ \mathop x\limits^ \cdot = 2y\left({b + c{x^2} + 2{y^2}} \right), \mathop y\limits^ \cdot = - 2x\left({a + 2{x^2} + c{y^2}} \right) \] under perturbations of polynomials with degree \(n\) is not more than \(3\left[ {\frac{{n - 1}}{4}} \right] + 12\left[ {\frac{{n - 3}}{4}} \right] + 22\) (taking into account the multiplicity), where \(a< 0, b< 0\) and \(c< - 2\).

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34D10 Perturbations of ordinary differential equations
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