Poincar´e’s Reversibility Condition

We consider a real planar analytic vector field, X, such that the origin, O, is a centre for the linearization of X. Poincar´e’s condition of reversibility with respect to a line passing through O is then a sufficient condition for O to be a centre for the vector field X. We provide necessary and sufficient conditions, involving the vanishing of certain polynomials in the coefficients in the expansion of X, for reversibility. We also show that if the linearization, LŽx., of the divergence of X is non-trivial, then the only possible reversibility line is given by LŽx. 0; in such cases, this provides the basis for a simple test of reversibility. We examine the consequences of our various tests for quadratic and cubic vector fields; all nonHamiltonian cases are discussed. When LŽx. 0 in cubic systems, it is possible for the reversibility line Žif it exists. to be unique, but it is also possible for there to be two such lines. These possibilities are characterized algebraically, and a prescription is provided for determining the reversibility lineŽs. in each case.