Let $f:\mathbb R^n\to\mathbb R$ be complete and $\dot x=f(x)$ the associated dynamic system with flow $\Phi: \mathbb R\times\mathbb R^n\to\mathbb R$. Then I want to show $\omega(x_0)=\mathcal{O}(x_0)$ if $x_0\in\mathbb R^n$ is peridodic (there is a $T>0$ such that $\Phi(T,x_0)=x_0$), where $\omega(x_0)$ is the limit set of $x_0$ and $\mathcal O(x_0)$ its orbit.
I have tried to show this via both inlusions but I couldn't make it far.
For $\subseteq$ I assume that some $y\in\omega(x_0)$. So there is a sequence $t_n\to\infty$ such that $\Phi(t_n,x_0)\to y$. Now I don't know how to continue properly. If $x_0$ is periodic, then maybe I can conclude that there is a $t_0$ such that $t_m\to t_0$ and $\Phi(t_0,x_0)=y$?
For $\supseteq$ I take a $y\in\mathcal O(x_0)$. Then there is a $t_0\in\mathbb R$ such that $\Phi(t_0,x_0)=y$. Now I take an arbitrary sequence $t_n\to t_0$, then $\Phi(t_n,x_0)\to y$. Since we have $\Phi(T+t_0,x_0)=y$ I can maybe build a sequence $t_n'\to\infty$ with $\Phi(t_n',x_0)\to y$. I was thinking about something like $t_n':=t_n+nT$.
"$\subseteq$": A hint: $\mathcal{O}(x_0)$ is compact (as the image of the compact space $\mathbb{R}/T \mathbb{Z}$ under the quotient map of $\Phi$), hence it is a closed subset of $\mathbb{R}^n$.
"$\supseteq$": For each $x \in \mathcal{O}(x_0)$ there holds $\lim\limits_{k \to \infty} \Phi(kT + \tau, x_0) = x$, where $\tau \in [0, T)$ is such that $\Phi(\tau, x_0) = x$.