Polynomial Differential Equations with Small coefficients
M.A.M. Alwash

TL;DR
This paper establishes bounds on polynomial differential equations' coefficients to determine the number of complex and real periodic solutions, proving a recent conjecture and applying results to limit cycle counts.
Contribution
It provides explicit bounds on coefficients ensuring a specific number of periodic solutions and proves a recent conjecture on their count.
Findings
Explicit bounds guarantee n complex periodic solutions.
Improved bounds for real periodic solutions in most classes.
Proof of a recent conjecture on the number of periodic solutions.
Abstract
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are 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 on 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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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Nonlinear Differential Equations Analysis · Numerical methods for differential equations
