Poincare'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. Poincare's condition of reversibi lity with respect to a line passing through O is then a sufficient conditio n for O to be a centre for the vector field X. We provide necessary and suf ficient conditions, involving the vanishing of certain polynomials in the c oefficients in the expansion of X, for reversibility. We also show that if the linearization, L(x), of the divergence of X is non-trivial, then the on ly possible reversibility line is given by L(x) = 0; in such cases, this pr ovides the basis for a simple test of reversibility. We examine the consequ ences of our various tests for quadratic and cubic vector fields; all non-H amiltonian cases are discussed. When L(x) equivalent to 0 in cubic systems, it is possible for the reversibility line (if it exists) to be unique, but it is also possible for there to be two such lines. These possibilities ar e characterized algebraically, and a prescription is provided for determini ng the reversibility line(s) in each case. (C) 2001 Academic Press.