Upper bounds for the number of zeroes for some Abelian integrals

Details Upper bounds for the number of zeroes for some Abelian integrals Upper bounds for the number of zeroes for some Abelian integrals

2010-12-23

arxiv.org/abs/1012.5201v1

Mathematics

Dynamical Systems

Scientific paper

Consider the vector field $x'= -yG(x, y), y'=xG(x, y),$ where the set of critical points $\{G(x, y) = 0\}$ is formed by $K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree $n$ and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of $K$ and $n.$ Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for $K\le4$ we recover or improve some results obtained in several previous works.

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