Periods of one dimensional periodic differential equations-II [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Armengol Gasull


Smooth non-autonomous $T$-periodic differential equations $x'(t)=f(t,x(t))$ defined in $RxK^n$, where $K$ is $R$ or $C$ and $n>1$ 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 periods of the non-constant periodic solutions of the scalar differential equation are divisors of the period of the equation, that is $T/S\in N.$ We also prove similar results in the one-dimensional holomorphic setting. A main difference in this situation is that the periods of the periodic solutions are commensurable with the period of the equation, that is $T/S\in Q.$ Moreover, in both cases, we study the structure of the set of the periods of all the periodic solutions of a given equation.

This talk is the second part of the one about the same subject given by Anna Cima some months ago. Nevertheless there is no need to have attended to the first one to follow it. I will focus mainly on the one-dimensional holomorphic setting. Finally, some open questions are discussed.

It is based on a joint work with Anna Cima and Francesc Mañosas