Let $f \in C^1(\mathbb{R}^n, \mathbb{R}^n)$. We observe the ODE $\dot{x} = f(x)$. Let $\Phi: \mathbb{R} \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ be its unique phase flow. We assume that for some $x_0 \in \mathbb{R}^n$, $n_1, n_2 \in \mathbb{N}$ and $T>0$ we have that $$ \Phi(n_1T, x_0) = \Phi(n_2T, x_0) = x_0. $$ Is it then true that $\Phi(\mathrm{gcd}(n_1, n_2)T, x_0) = x_0$ ($\mathrm{gcd}$ is the greatest common divider) or even $\Phi(T, x_0) = x_0$?
I really believe that both are true... . If I can complete a period in $n_1T$ (let's assume that $n_2 > n_1$) then I can complete a period in $(n_2-n_1)T$. Is this correct and if yes, what exactly are the implications of this thought process?