Periods of solutions of periodic differential equations

Abstract

Smooth non-autonomous $T$-periodic differential equations $x'(t)=f(t,x(t))$ defined in $ \mathbb R \times \mathbb K ^n$, where $ \mathbb K $ is $ \mathbb R $ or $\mathbb C$ and $n\ge 2$ 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 $\mathcal{C}^1$-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 $T/S\in \mathbb N .$ 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 $T/S\in \mathbb Q .$