Polynomial differential equations with small coefficients
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.
Received: January 2009; Revised: May 2009; Published: September 2009.
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