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 + n k=1 P2k+1(x, y), dy dt =−x + n k=1 Q2k+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), Q2k+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. © 2005 Elsevier Inc. All rights reserved