Electron. J. Diff. Eqns., Vol. 2003(2003), No. 34, pp. 1-7.

Centering conditions for planar septic systems

Evgenii P. Volokitin

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.

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Evgenii P. Volokitin
Sobolev Institute of Mathematics
Novosibirsk, 630090, Russia
email: volok@math.nsc.ru

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