Geodesic
Constant invariant solutions of the Poincaré center-focus problem
Electronic journal of differential equations, Tome 2010 (2010)
We consider the classical Poincaré problem dx dy = - y - $p(x, y), = x + q(x, y)$ dt dt where p, q are homogeneous polynomials of degree n $\geq 2$. Associated with this system is an Abel differential equation d$\rho = \psi 3\rho 3 + \psi 2\rho 2$ d$\theta $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 Poincaré problem if n $\geq 9$ is an odd integer. We also show that both classes of solutions produce polynomial solutions to the problem.
Classification : 34A05, 34C25
Keywords: center-focus problem, Abel differential equation, constant invariant, symmetric centers 
@article{EJDE_2010__2010__a84,
     author = {Nicklason,  Gary R.},
     title = {Constant invariant solutions of the {Poincar\'e} center-focus problem},
     journal = {Electronic journal of differential equations},
     year = {2010},
     volume = {2010},
     zbl = {1218.34033},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a84/}
}
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Nicklason,  Gary R. Constant invariant solutions of the Poincaré center-focus problem. Electronic journal of differential equations, Tome 2010 (2010). http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a84/