Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 130, pp. 1-11.
Title: Constant invariant solutions of the Poincare center-focus problem
Author: Gary R. Nicklason (Cape Breton Univ., Sydney, Nova Scotia, Canada)
Abstract:
We consider the classical Poincare problem
$$
\frac{dx}{dt}=-y-p(x,y),\quad \frac{dy}{dt}=x+q(x,y)
$$
where $p,q$ are homogeneous polynomials of degree $n \geq 2$.
Associated with this system is an Abel differential equation
$$
\frac{d\rho}{d\theta}=\psi_3\rho^3 + \psi_2\rho^2
$$
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 $n \geq 9$ is an odd integer.
We also show that both classes of solutions produce polynomial
solutions to the problem.
Submitted June 7, 2010. Published September 14, 2010.
Math Subject Classifications: 34A05, 34C25.
Key Words: Center-focus problem; Abel differential equation;
constant invariant; symmetric centers.