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Limit cycles in a quartic system with a third-order nilpotent singular point. (English) Zbl 1446.34052

Summary: In this paper, limit cycles bifurcating from a third-order nilpotent critical point in a class of quartic planar systems are studied. With the aid of computer algebra system MAPLE, the first 12 Lyapunov constants are deduced by the normal form method. As a result, sufficient and necessary center conditions are derived, and the fact that there exist 12 or 13 limit cycles bifurcating from the nilpotent critical point is proved by different perturbations. The result in [J. Qiu and F. Li, ibid. 2015, Paper No. 29, 10 p. (2015; Zbl 1361.34028)] is improved.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for 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
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C25 Periodic solutions to ordinary differential equations

Citations:

Zbl 1361.34028

Software:

Maple
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Full Text: DOI

References:

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