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

In this paper we investigate the problem of center-focus for the systemdxdt=y+∑k=1nP2k+1(x,y),dydt=−x+∑k=1nQ2k+1(x,y) which is regarded as a perturbed one of a planar linear system dxdt=y, dydt=−x (where P2k+1(x,y)P2k+1(x,y), Q2k+1(x,y)Q2k+1(x,y), k=1,2,…,nk=1,2,…,n, are (2k+1)(2k+1)th-degree homogeneous polynomials in (x,y)(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.