Electron. J. Diff. Equ., Vol. 2015 (2015), No. 189, pp. 1-17.

An Abel type cubic system

Gary R. Nicklason

We consider center conditions for plane polynomial systems of Abel type consisting of a linear center perturbed by the sum of 2 homogeneous polynomials of degrees n and 2n-1 where $n \ge 2$. Using properties of Abel equations we obtain two general systems valid for arbitrary values on n. For the cubic n=2 systems we find several sets of new center conditions, some of which show that the results in a paper by Hill, Lloyd and Pearson which were conjectured to be complete are in fact not complete. We also present a particular system which appears to be a counterexample to a conjecture by Zoladek et al. regarding rational reversibility in cubic polynomial systems.

Submitted March 30, 2015. Published July 16, 2015.
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, B1P 6L2, Canada
email: gary_nicklason@cbu.ca

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