I'm doing a project on a chaotic periodically driven system, and I would like to construct a Poincare map of the system. Everywhere says that for a periodically driven system, you simply choose you Poincare section by samplying the position and velocity of the system with a period equal to that of the driving period.
My question is this: what poincare section does this correspond to? All other examples I've seen of Poincare sections consist of a set of values in the phase space. E.g., if someone is looking at the 3D Lorenz attractor, they might take a poincare section of z=0, or x=0, etc. I've haven't seen anyone mention taking a Poincare system with respect to time. What is the meaning of a Poincare section at repeated time intervals? Thanks!
The procedure that you have described is also called a stroboscopic map. It is strongly connected to the following property of $T$-periodic dynamical systems $\dot{x} = f(x, t)$: if $\gamma(t)$ is a solution then $\gamma(t+kT), \, k \in \mathbb{Z}$ is a solution too. I'll try to sketch the idea why this stroboscopic map is a Poincaré map in disguise. The idea is simple, but the full and rigorous proof requires a bit of topology (quotient spaces, covering spaces, vector fields, projections, ...).
Since the system is non-autonomous, you have to consider so called extended phase space: \begin{equation} \label{eq} \dot{x} = f(x, \tau), \, \dot{\tau} = 1.\;\;\;\;\;\;\;\;\;\; (\ast) \end{equation}
If your phase space was a manifold $\mathcal{M}$, then an extended phase space is a $\mathcal{M} \times \mathbb{R}$. In this extended phase space integral curves don't intersect (that's the whole point of passing to it in non-autonomous case). Note that all trajectories always go through $\tau = kT$, $k \in \mathbb{Z}$ because $\tau$ changes linearly in time. Let's make a quotient space out of $\mathcal{M} \times \mathbb{R}$ by choosing $(x, \tau) \sim (x, \tau + kT)$ as an a equivalence (periodicity of the vector field strongly suggests to do this). The quotient space is homeomorphic to $\mathcal{M} \times \mathbb{S}^1$ and we can project points from $\mathcal{M} \times \mathbb{R}$ to $\mathcal{M} \times \mathbb{S}^1$ via mapping $(x,\, \tau) \mapsto (x,\, \tau\mod T)$. What will happen if we take an integral curve of system $(\ast)$ from the extended phase space and project it on quotient space? The projection of curve $(\gamma(t), t)$ will keep returning to $\mathcal{M} \times \lbrace 0 \rbrace$ and it will intersect $\mathcal{M} \times \{0\}$ at points $\{ \gamma(kT) \}_{k \in \mathbb{Z}}$. So stroboscopic map is a Poincaré map of the quotient system.