## Differential and Integral Equations

### 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 $T/S\in \mathbb Q .$

#### Article information

Source
Differential Integral Equations, Volume 29, Number 9/10 (2016), 905-922.

Dates
First available in Project Euclid: 14 June 2016

https://projecteuclid.org/euclid.die/1465912609

Mathematical Reviews number (MathSciNet)
MR3513586

Zentralblatt MATH identifier
06644054

#### Citation

Cima, Anna; Gasull, Armengo; Mañosas, Francesc. Periods of solutions of periodic differential equations. Differential Integral Equations 29 (2016), no. 9/10, 905--922. https://projecteuclid.org/euclid.die/1465912609