Open Access
2016 Homogeneous-Like Generalized Cubic Systems
G. R. Nicklason
Int. J. Differ. Equ. 2016: 1-15 (2016). DOI: 10.1155/2016/7640340

Abstract

We consider properties and center conditions for plane polynomial systems of the forms x˙=-y-p1(x,y)-p2(x,y), y˙=x+q1(x,y)+q2(x,y) where p1, q1 and p2, q2 are polynomials of degrees n and 2n-1, respectively, for integers n2. We restrict our attention to those systems for which yp2(x,y)+xq2(x,y)=0. In this case the system can be transformed to a trigonometric Abel equation which is similar in form to the one obtained for homogeneous systems (p2=q2=0). From this we show that any center condition of a homogeneous system for a given n can be transformed to a center condition of the corresponding generalized cubic system and we use a similar idea to obtain center conditions for several other related systems. As in the case of the homogeneous system, these systems can also be transformed to Abel equations having rational coefficients and we briefly discuss an application of this to a particular Abel equation.

Citation

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G. R. Nicklason. "Homogeneous-Like Generalized Cubic Systems." Int. J. Differ. Equ. 2016 1 - 15, 2016. https://doi.org/10.1155/2016/7640340

Information

Received: 20 April 2016; Accepted: 25 July 2016; Published: 2016
First available in Project Euclid: 21 December 2016

zbMATH: 1357.34064
MathSciNet: MR3548418
Digital Object Identifier: 10.1155/2016/7640340

Rights: Copyright © 2016 Hindawi

Vol.2016 • 2016
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