The center-focus problem and reversibility

In this paper we discuss the relation between time-reversibility and the ce nter/focus problem. We show the following: for any analytic planar system X , if j(1)X, the one-jet of X, is conjugated to (-y, x) then X is analytical ly time-reversible if and only if it is a center; if j(1)X is conjugated to (y, 0), then some sufficient and necessary conditions for X to have a cent er are given. In particular, the reversibility of certain types of polynomi al vector Fields is studied. (C) 2001 Academic Press.