Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 34, pp. 1-7.
Title: Centering conditions for planar septic systems
Author: Evgenii P. Volokitin (Sobolev Inst. of Math., Novosibirsk, Russia)
Abstract:
We find centering conditions for the following $O$-symmetric system
of degree 7:
$$\displaylines{
\dot x=y+ x (H_2 (x,y)+H_6 (x,y)),\cr
\dot y=-x+ y (H_2 (x,y)+H_6 (x,y)),
}$$
where $H_2 (x,y)$ and $H_6 (x,y)$ are homogeneous polynomials
of degrees 2 and 6, respectively.
In some cases, we can find commuting systems and first integrals for the
original system. We also study the geometry of the central region.
Submitted August 22, 2002. Published April 3, 2003.
Math Subject Classifications: 34C05, 34C25.
Key Words: centering conditions; isochronicity; commutativity.