Abstract
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 [Qiu et al. in Adv. Differ. Equ. 2015(1):1, 2015] is improved.
Highlights
An isolated critical point O(0, 0) is called a nilpotent singular point if the linear part of systems has double zero eigenvalues but the matrixes of the linearized systems at the origin are not identically null
More and more attention has been paid to the center problem and bifurcation of limit cycles of a system with a nilpotent critical point recently
3.1 Perturbation method of small parameters At first, we will prove that the perturbed system of (1.7) can generate twelve limit cycles enclosing an elementary node at the origin of unperturbed system (1.7) when the third-order nilpotent critical point O(0, 0) is a 12th-order weak focus by perturbing the coefficients
Summary
An isolated critical point O(0, 0) is called a nilpotent singular point if the linear part of systems has double zero eigenvalues but the matrixes of the linearized systems at the origin are not identically null. More and more attention has been paid to the center problem and bifurcation of limit cycles of a system with a nilpotent critical point recently. For quadratic and cubic Hamiltonian systems, they obtained necessary and sufficient conditions for a nilpotent critical point to be a center, a cusp, or a saddle.
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