Wikipedia's article about the kicked rotator says that it's Hamiltonian is \begin{equation} H(p,q,t)=\frac{1}{2}p^2+K\cos(q)\sum_{n=-\infty}^{\infty}\delta(t-n) \end{equation} and it's Poincaré surface of section is \begin{align} q_{n+1}&=q_n+p_{n+1} \\ p_{n+1}&=p_n+K\sin(q_n). \end{align}
I solved Hamilton equations and got \begin{equation} \dot{q}=p, \qquad \dot{p}=K\sin(q)\sum_{n=-\infty}^{\infty}\delta(t-n). \end{equation} How do I construct this Poincaré surface of section to get the discrete equations from the continuous ones?
What is the difference between this and the standard map? (In the standard map I need to take $p$ and $q$ modulo $2\pi$ and in the kicked rotator only the $q$ is modulo $2\pi$?)
And how can I construct its phase space plot? (Is it possible to do this in LaTeX?)
Thanks