The criterion of the center-focus for a class of planar perturbed polynomial systems with (2n+1)th degree

In this paper we investigate the problem of center-focus for the system dx/dt = y + Sigma(n)(k=1) P2k+1 (x, y), dy/dt = -x + Sigma(n)(k=1) Q(2k+1) (x, y) which is regarded as a perturbed one of a planar linear system dx/dt = y, dy/dt = -x (where P2k+1(x, y), Q(2k+1)(x, y), k = 1, 2,..., n, are (2k + 1)th-degree homogeneous polynomials in (x, y)). We shall give a simple and convenient method which can immediately distinguish that the singular point O is a center or fine focus and the stability of the singular point can be determined by the matrices consist of the coefficients of perturbed terms. (c) 2005 Elsevier Inc. All rights reserved.