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.

  • $\begingroup$ @nmasanta : why did you edited my Q1) in $Q.1)$ ? $\endgroup$ – 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.


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