Dynamical system with periodic orbit

Question

We are given dynamical system $\phi $ in $R^2$, and know that it has periodic orbit (means $\phi(T,x_0)=x_0$ for some $T>0$ and $x_0 \in R$). We are asked to prove that the system has stationar point (means $\phi(t,x)=x$ for some $x \in R^2$ and all $t \in R$. I appriciate any kind of help.

Answer 1

Let $\gamma(t) = \phi(t, x_0)$, $t\in [0, T]$ be a curve. We choose $T$ small so that $\gamma$ is a simple closed curve. Then $\gamma$ bounds a region $U$ in $\mathbb R^2$ (so that $\partial U$ is the curve $\gamma$).

First of all, if $y\in \bar U$ then $\phi(t, y) \in \bar U$ for any $t$ (why?). So we can restrict our mapping to $\phi(t, \cdot) :\bar U \to \bar U$.

Second, as $\bar U$ is homeomorphic to a disk, all $\phi(t, \cdot)$ has a fixed point in $\bar U$. Thus we can apply the general result in:

Compact space, continuous dynamical system, stationary point

to claim that there is a stationary point.

(I believe that there are easier proof...)