Upper bounds for the number of zeroes for some Abelian integrals

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 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 on 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.Peer ReviewedPostprint (published version

Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

“…The first one, in Section 3, deals with polynomial expressions with two square roots of polynomials. These equations arise in the study of the number of zeros for some Abelian integrals (see [15]). The second one, in Section 4, handles with central configuration problems in Celestial Mechanics.…”

Poincaré Bifurcations of Two Classes of Polynomial Systems

“…In [12], the authors studied the number of limit cycles of system (1) with ( , ) = ( + )( + ) and obtain that the system can have at most 3[( − 1)/2] + 2 limit cycles if ̸ = and 2[( − 1)/2] + 1 if = , respectively. In [13], the authors studied the case the curves ( , ) = 0 are three lines, two of them parallel and one perpendicular, and [14,15] studied the case the curves are ( > 3) lines, and any two of them are parallel or perpendicular directions. The authors in [16] studied the case the curves are consistent by nonzero points.…”

Limit cycles appearing from the perturbation of a system with a multiple line of critical points

“…The cases where the curve {C(x, y) = 0} is one line; two parallel lines; two orthogonal lines; k lines, parallel to two orthogonal directions; or k isolated points are some of these situations, see [1,3,4,6,7], respectively. Notice that none of these algebraic curves has a multiple factor.…”