Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center

Abstract

In this paper, we deal with the following differential systeṁwhere p, q are positive integers, and P (x, y), Q(x, y) are real polynomials of degree n, we obtain an upper bound for the maximum number of limit cycles bifurcating from the period annulus of a quasi-homogeneous center, that is (n − 1)p 1 + (t + 1)q − 1 + 2rp 1 q 1 (q + 3) + 2tqrp 1 q 1 , where t = [n/2q] + 2, (p, q) = r(p 1 , q 1 ), p 1 and q 1 are coprime.

Planar Semi-quasi Homogeneous Polynomial Differential Systems with a Given Degree

“…The quasi-homogeneous polynomial differential systems have also gained wide attention since the beginning of the 21st century. The result obtained mainly include the integrability [3,4,16,20,21,27,31], polynomial and rational first integrability [2,9], normal forms [4], centers [1,5,6,35,42,43] and limit cycles [22,23,25,29]. Recently, the authors of [20] establish an algorithm for obtaining all the quasi-homogeneous but non-homogeneous polynomial systems with a given degree.…”

An algorithm for providing the normal forms of spatial quasi-homogeneous polynomial differential systems

“…(4) It is a QH system with weight vector (2, 1, 1, 3), as can be seen from (2). But this system is not maximal, because it can be completed to ẋ = xyz + y 3 + x 2 , ẏ = y 2 z + xz 2 + xy, ż = yz 2 + y 2 + xz, (5) which still is QH with the weight vector (3, 2, 1, 4).…”

“…In Liang-Huang-Zhao [10] are studied the phase portraits. The centers and limit cycles are discussed in Tang-Wang-Zhang [13], Geng-Lian [5], Li-Wu [9] and Xiong-Han-Wang [14]. Chiba [2] and Yoshida [15] have explored the Kowalevski exponents.…”