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. © 2012 Elsevier Ltd. All rights reserved.
|Journal||Nonlinear Analysis, Theory, Methods and Applications|
|Publication status||Published - 1 Sep 2012|
- Abelian integrals
- Chebyshev system
- Limit cycles
- Number of zeroes of real functions
- Weak 16th Hilbert's Problem