Periods of solutions of periodic differential equations

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Abstract

Smooth non-Autonomous T-periodic differential equations x0(t) = f(t; x(t)) defined in ℝ × Kn, where K is ℝ or C and n ≥ 2 can have periodic solutions with any arbitrary period S. We show that this is not the case when n = 1: We prove that in the real C1-setting the period of a non-constant periodic solution of the scalar differential equation is a divisor of the period of the equation, that is T=SN: Moreover, we characterize the structure of the set of the periods of all the periodic solutions of a given equation. We also prove similar results in the one-dimensional holomorphic setting. In this situation the period of any non-constant periodic solution is commensurable with the period of the equation, that is T=Sℚ:.
Original languageEnglish
Pages (from-to)905-922
JournalDifferential and Integral Equations
Volume29
Issue number9-10
Publication statusPublished - 1 Jan 2016

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