Differential and Integral Equations

Periods of solutions of periodic differential equations

Anna Cima, Armengo Gasull, and Francesc Mañosas

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 14 June 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C25: Periodic solutions 37C27: Periodic orbits of vector fields and flows 37G15: Bifurcations of limit cycles and periodic orbits


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.

Export citation