My lectures examine in detail Poincare’s map for dynamical systems with a periodic trajectory. They, however, mention only in passing the construction for a dynamical system with a periodic phase space, as follows.
Assume $v\in C^r$ be a $n$-times smooth periodic vector field and assume $$\dot x=v(t,x),\quad t\in\mathbb R, x\in\mathbb R^n, \quad v(t+T,x)=v(t,x),\quad T=\frac{2\pi}{\omega}>0$$
Let us now write the system as an autonomous one $$\theta:\mathbb R^1\to S^1:t\to\theta(t)=\omega t(\mod 2\pi)\quad (S^1\text{is the unit circle})$$
$$\dot x=v(\theta,x)$$ $$\dot\theta=\omega (\mod 2\pi)$$ Let us denote the flow by $g^t=\{x(t),\theta(t)=\omega t+\theta_0(\mod 2\pi)\}$
Now define the section $$\Gamma^\overline{\theta_0}=\{(x,\theta)\in\mathbb R\times S^1:\theta=\overline{\theta_0}\}$$ $\Gamma^\overline{\theta_0}$ is a global section since $(0,1)$ is normal to it.
Finally consider Poincare's map defined by $$P:\Gamma^\overline{\theta_0}\to\Gamma^\overline{\theta_0}:(x(\frac{\overline{\theta_0}-\theta_0}{\omega}),\overline{\theta_0})\to (x(\frac{\overline{\theta_0}-\theta_0+2\pi}{\omega}),\overline{\theta_0}+2\pi)$$ or $$x(\frac{\overline{\theta_0}-\theta_0}{\omega})\to x(\frac{\overline{\theta_0}-\theta_0+2\pi}{\omega})$$
I am looking for a proof of what I believe is the following
Theorem: Assume $x_0,x_1$ are two distinct points on an orbit in the periodic vector space. Construct the sections $\Gamma^\overline{\theta_0},\Gamma^\overline{\theta_1}$ as above and define their respective Poincare’s maps $P_0,P_1$. Then $P_0,P_1$ are $C^r$ conjugate, i.e. there exits a homeomorphism $h\in C^r$, such that $$P_1 \circ h=h\circ P_0$$
p.s. I suspect I haven't formulated the right theorem, or haven't formulated any theorem correctly. I will do as soon as anyone corrects me.
p.p.s. An online resource is just as welcome as a proof in an answer. Any comments would be appreciated as well as references to books that are free to download or read online.
You can take $h$ just to be the flow of the ODE from the section $\Gamma^{\theta_0}$ to $\Gamma^{\theta_1}$ and the conjugacy follows. Agree?