New lower bound for the local cyclicity of quintic planar polynomial vector fields [ Back ]

15:30 to 16:30
UAB - Dept. Matemàtiques (C1/-128)
Luiz Fernando Gouveia
Universitat Autònoma de Barcelona


One of problems that remains open is the second part of the 16th Hilbert's problem that consist in to determine the maximal number (named $H(n)$) of limit cycles, and their relative positions, of a planar polynomial systems of degree $n$. We are interested here in the local version of this Hilbert's 16th problem, that is, to provide the number $M(n)$ of small amplitude limit cycles bifurcating from a point of center-focus type. Clearly $M(n)\leq H(n).$

Bautin (1954) proved that $M(2) = 3$. Sibirskii (1965) proved that for cubic systems without quadratic terms have no more than five limit cycles bifurcating from one critical point. Zoladek (1995, 2015) found an example where eleven limit cycles could be bifurcated from a single critical point of a cubic system and Christopher (2005), studying only the Taylor developments of the Liapunov quantities, provides a simpler proof of the existence of a cubic system with 11 limit cycles.

We prove that $M(5)\geq 33$. More concretely, we present a center such that 33 limit cycles bifurcate from the origin. We remark that this lower bound coincide with the value, $M(n)=n^2+3n-7$. The computations have been done using a generalization of the parallelization procedure, introduced by Liang and Torregrosa, for finding the higher order terms in the perturbation parameters of the coefficients of the return map.

This is a joint work with Joan Torregrosa.

The talk will be in Spanish.