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The cyclicity of period annuli of a Hamilton system with nilpotent saddle points. (Chinese. English summary) Zbl 1449.34109

Summary: In this paper, we study the cyclicity of period annuli of the following cubic Hamilton system with nilpotent saddle points \[\frac{{\mathrm{d}}x}{{\mathrm{d}}t} = 4{x^2}y + 4{y^3} - y,\; \frac{{\mathrm{d}}y}{{\mathrm{d}}t} = 4{x^2} - 4x{y^2} + x.\] By using the first order Melnikov function and Picard-Fuchs equation, we obtain that the above system under perturbations of real polynomials with degree \(n\) can bifurcate at most \(4n + 10\) limit cycles (taking into account the multiplicity).

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
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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