Polynomial differential equations with small coefficients

M. A. M. Alwash

doi:10.3934/dcds.2009.25.1129

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2009, Volume 25, Issue 4: 1129-1141. Doi: 10.3934/dcds.2009.25.1129

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Polynomial differential equations with small coefficients

M. A. M. Alwash^1,

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

Received: December 31, 2008

Revised: April 30, 2009

Published: October 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 34C25, 34C07, 34C05; Secondary: 37G15.

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