Electron. J. Diff. Eqns., Vol. 2007(2007), No. 101, pp. 1-15.

Complex centers of polynomial differential equations

Mohamad Ali M. Alwash

We present some results on the existence and nonexistence of centers for polynomial first order ordinary differential equations with complex coefficients. In particular, we show that binomial differential equations without linear terms do not have complex centers. Classes of polynomial differential equations, with more than two terms, are presented that do not have complex centers. We also study the relation between complex centers and the Pugh problem. An algorithm is described to solve the Pugh problem for equations without complex centers. The method of proof involves phase plane analysis of the polar equations and a local study of periodic solutions.

Submitted March 19, 2007. Published July 25, 2007.
Math Subject Classifications: 34C05, 34C07, 34C25, 37C10, 13P10.
Key Words: Polynomial differential equations; periodic solutions; multiplicity; centers; Pugh problem; Groebner bases.

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Mohamad Ali M. Alwash
Department of Mathematics,
West Los Angeles College and University of California, Los Angeles
9000 0verland Avenue, Los Angeles, CA 90230-3519, USA
email: alwash@math.ucla.edu and alwashm@wlac.edu

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