The Poincare center-focus problem for a class of higher order polynomial differential systems

Abstract

In this paper, I have proved that for a class of polynomial differential systems of degree n+1 ( where n is an arbitrary positive integer) the composition conjecture is true. I give the sufficient and necessary conditions for these differential systems to have a center at origin point by using a different method from the previous references. By this I can obtain all the focal values of these systems for an arbitrary n and their expressions are succinct and beautiful. I believe that the idea and method of this article can be used to solve the center-focus problem of more high-order polynomial differential systems.