Electron. J. Diff. Equ., Vol. 2010(2010), No. 130, pp. 1-11.

Constant invariant solutions of the Poincare center-focus problem

Gary R. Nicklason

Abstrac: 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.

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Gary R. Nicklason
Mathematics, Physics and Geology
Cape Breton University
Sydney, Nova Scotia, Canada B1P 6L2
email: gary_nicklason@capebretonu.ca

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