Differential and Integral Equations

Periods of solutions of periodic differential equations

Anna Cima, Armengo Gasull, and Francesc Mañosas

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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

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