 Hyperbolic canonical chart for a quadratic oscillator

For $p>0$, $Q=\sqrt{2p}\sinh q$ and $P=\sqrt{2p}\cosh q$ satisfy $\{Q,P\}=1$. They transform $p\cosh 2q$ into $(Q^2+P^2)/2$. Their inverse is $p=(P^2-Q^2)/2$ and $q=\mathrm{artanh}(Q/P)$, so the chart covers only $P>|Q|$, not the whole oscillator plane. Oscillator evolution beyond that wedge is not a global real continuation of the original variables.