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In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21] , to prove that there exists a family of cubic centers, labeled by C D 31 12 in [25] , with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields. [ABSTRACT FROM AUTHOR]

We are interested in small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point, that we locate at the origin. We develop a parallelization technique to study higher order developments, with respect to the parameters, of the return map near the origin. This technique is useful to study lower bounds for the local cyclicity of centers. We denote by M (n) the maximum number of limit cycles bifurcating from the origin via a degenerate Hopf bifurcation for a polynomial vector field of degree n. We get lower bounds for the local cyclicity of some known cubic centers and we prove that M (4) ≥ 20 , M (5) ≥ 33 , M (7) ≥ 61 , M (8) ≥ 76 , and M (9) ≥ 88. [ABSTRACT FROM AUTHOR]

Abstract: We classify all centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4. [Copyright &y& Elsevier]

Abstract: This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where is a prime p or a power of a prime , and show that these systems can have a focus order ; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too. [Copyright &y& Elsevier]