Integrability, degenerate centers, and limit cycles for a class of polynomial differential systems

摘要

We consider the class of polynomial differential equations x = P-n(x, y) + Pn+1(x, y) + Pn+2(x, y), y = Q(n)(x, y) + Q(n+1)(x, y) + Q(n+2)(x, y), for n >= 1 and where P-i and Q(i) are homogeneous polynomials of degree i. These systems have a linearly zero singular point at the origin if n >= 2. Inside this class, we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center, i.e., a center with linear part identically zero. Moreover, under additional conditions such Darboux integrable systems can have at most one limit cycle. We provide the explicit expression of this limit cycle. (c) 2006 Elsevier Ltd. All rights reserved.