TY - JOUR
T1 - Upper bounds for the number of zeroes for some Abelian integrals
AU - Gasull, Armengol
AU - Tomás Lázaro, J.
AU - Torregrosa, Joan
PY - 2012/9/1
Y1 - 2012/9/1
N2 - 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.
AB - 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.
KW - Abelian integrals
KW - Chebyshev system
KW - Limit cycles
KW - Number of zeroes of real functions
KW - Weak 16th Hilbert's Problem
UR - https://www.scopus.com/pages/publications/84862115280
U2 - 10.1016/j.na.2012.04.033
DO - 10.1016/j.na.2012.04.033
M3 - Article
SN - 0362-546X
VL - 75
SP - 5169
EP - 5179
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
IS - 13
ER -