Limit Cycles Coming from Some Uniform Isochronous Centers

作者：Liang, Haihua; Llibre, Jaume; Torregrosa, Joan^*

来源：Advanced Nonlinear Studies, 2016, 16(2): 197-220.

DOI：10.1515/ans-2015-5010

• Periodic Solution
• Uniform Isochronous Centers
• Averaging Theory
• Weak Hilbert Problem

This article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers @@@ (x) over dot = -y + x(2)y(x(2) + y(2))(n), (y) over dot = x + xy(2) (x(2) + y(2))n, @@@ of degree 2n + 3 and we perturb them inside the class of all polynomial differential systems of degree 2n + 3. For n = 0, 1 we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n = 2 we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center.

• 出版日期2016-5
• 单位广东技术师范大学