Polynomial differential equations with small coefficients

Pages: 1129 - 1141, Volume 25, Issue 4, December 2009

doi:10.3934/dcds.2009.25.1129       Abstract        Full Text (164.7K)       Related Articles

M. A. M. Alwash - Department of Mathematics, West Los Angeles College, Los Angeles, CA 90230-3519, United States (email)

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.

Keywords:  Periodic solutions, limit cycles, polynomial differential equations, Abel differential equation, Hilbert's sixteenth problem.
Mathematics Subject Classification:  Primary: 34C25, 34C07, 34C05; Secondary: 37G15.

Received: January 2009;      Revised: May 2009;      Published: September 2009.