Upper bounds for the number of zeroes for some Abelian integralsReport as inadecuate



 Upper bounds for the number of zeroes for some Abelian integrals


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Consider the vector field $x= -yGx, y, y=xGx, y,$ where the set of critical points $\{Gx, y = 0\}$ 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.



Author: Armengol Gasull; J. Tomás Lázaro; Joan Torregrosa

Source: https://archive.org/







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