An Improvement on the Number of Limit Cycles Bifurcating from a Nondegenerate Center of Homogeneous Polynomial Systems

Abstract

In the two articles in Appl. Math. Comput., J. Giné [2012a, 2012b] studied the number of small limit cycles bifurcating from the origin of the system: [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are homogeneous polynomials of degree [Formula: see text]. It is shown that the maximal number of the small limit cycles, denoted by [Formula: see text], satisfies [Formula: see text] for [Formula: see text]; and [Formula: see text], [Formula: see text]. It seems that the correct answer for their case [Formula: see text] should be [Formula: see text]. In this paper, we apply Hopf bifurcation theory and normal form computation, and perturb the isolated, nondegenerate center (the origin) to prove that [Formula: see text] for [Formula: see text]; and [Formula: see text] for [Formula: see text], which improve Giné’s results with one more limit cycle for each case.