Restricted independence in displacement function for better estimation of cyclicity

Abstract

Since the independence of focal values is a sufficient condition to give a number of limit cycles arising from a center-focus equilibrium, in this paper we consider a restricted independence to a parametric curve, which gives a method not only to increase the lower bound for the cyclicity of the centerfocus equilibrium but also to be available when those focal values are not independent. We apply the method to a nondegenerate cubic center-focus variety and prove that the cyclicity reaches its an upper bound.

Bi-center problem and Hopf cyclicity of a Cubic Liénard system

“…, which implies that at most 2 small amplitude limit cycles bifurcate from center O by [2]. On the other hand, we have rank…”

“…Here A i s (i = 0, 1, 2, 3) are regarded as new independent parameters because of the arbitrariness of c i s (i = 0, 1, 2, 3). We consider the bi-center problem of system (2). Straight computations show that the Jacobi matrices of the vector field in system (2) at (0, 0), (1, 0), (a, 0) are…”

“…This implies that neither (0, 0) nor (a, 0) is a center or focus when c < 0. Thus, we only need to consider c > 0 for the bi-center problem of system (2). Further, by the transformation…”