Constant Invariant Solutions of the Poincare Center-Focus Problem

Abstract

We consider the classical Poincare problem dx/dt = -y - p(x, y), dy/dt = x + q(x, y) where p, q are homogeneous polynomials of degree n >= 2. Associated with this system is an Abel differential equation d rho/d theta = psi(3)rho(3) + psi(2)rho(2) in which the coefficients are trigonometric polynomials. We investigate two separate conditions which produce a constant first absolute invariant of this equation. One of these conditions leads to a new class of integrable, center conditions for the Poincare problem if n >= 9 is an odd integer. We also show that both classes of solutions produce polynomial solutions to the problem.