Polynomial differential equations with small coefficients

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December 2009, 25(4): 1129-1141. doi: 10.3934/dcds.2009.25.1129

Polynomial differential equations with small coefficients

1.	Department of Mathematics, West Los Angeles College, Los Angeles, CA 90230-3519, United States

Received January 2009 Revised May 2009 Published September 2009

Abstract
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Classes of polynomial non-autonomous differential equations of degree $n$ are considered. An explicit bound on the size of the coefficients is given which implies that each equation in the class has exactly $n$ complex periodic solutions. In most of the classes the upper bound can be improved when we consider real periodic solutions. We present a proof to a recent conjecture about the number of periodic solutions. The results are used to give upper bounds for the number of limit cycles of polynomial two-dimensional systems.

Keywords: Periodic solutions, Abel differential equation, limit cycles, polynomial differential equations, Hilbert's sixteenth problem..

Mathematics Subject Classification: Primary: 34C25, 34C07, 34C05; Secondary: 37G1.

Citation: M. A. M. Alwash. Polynomial differential equations with small coefficients. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1129-1141. doi: 10.3934/dcds.2009.25.1129

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