# Periodic orbit doesn't necessarily have periodic parametrization?

Let $$\dot x_t=f(x_t)$$ an ODE $$(x\in \mathbb R^n)$$.

Q1) What does mean : there exist a periodic orbit of the flow $$\Phi^t$$ associated to the ODE with least period $$T$$ ?

Q2) Let $$\Phi^t(x_0)$$ a periodic flow of the ODE. Let $$\Gamma _{x_0}=\{\varphi ^t(x_0)\mid t\geq 0\}.$$ Then $$\Gamma _{x_0}$$ is a periodic orbit. Is it possible to have a non periodic flow that lie on $$\Gamma _{x_0}$$ ?

For example, I'm thinking to the unit circle $$\{(\cos(t),\sin(t))\mid t\in [0,2\pi]\}=:\mathcal C$$. Would it be possible to have $$x(t)=(\cos(t^2),\sin(t^2))$$ where $$t\geq 0$$. This typically lies on $$\mathcal C$$ but it's not periodic.

• @nmasanta : why did you edited my Q1) in $Q.1)$ ? – user659895 Jun 29 at 7:56

Q1) There is $$x_0$$ s.t. $$\Phi^t (x_0)$$ is periodic.
Q2) Yes, if $$\Gamma _{x_0}$$ is not generated by a periodic solution. But as far as there is a periodic solution that lies on $$\Gamma _{x_0}$$, then the answer is no.