Upper bounds for the number of zeroes for some Abelian Integrals

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Document typeResearch report
Defense date2012-01-12
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Abstract
Abstract. Consider the vector field x0 = -yG(x, y), y0 = 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 4 we recover or
improve some results obtained in several previous works.
CitationLázaro, J.; Gasull, A.; Torregrosa, J. "Upper bounds for the number of zeroes for some Abelian Integrals". 2012.
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