The qualitative behavior of a second-order system with zero diagonal coefficient

This paper investigates the qualitative behavior of solutions of the autonomous planar system with zero diagonal coefficient ˙x = p2(y)q2(x)y, ˙ y = p3(y)q3(x)x + p4(y)q4(x)y. Under suitable assumptions, the necessary and sufficient conditions for all solutions to be oscillatory, and for the origin to be a global center are established. The theorems on the existence and uniqueness of nontrivial periodic solutions are also proved.  2003 Elsevier Inc. All rights reserved