Upper bounds for the number of zeroes for some Abelian integrals

2010/12/23 by Gasull, Armengol, Lázaro, J. Tomás, Torregrosa, Joan
#34C07 #34C08 #34C23 #37C27 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1012.5201

Abstract

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≤4 we recover or improve some results obtained in several previous works.

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