The Poincaré Map is the map $\Phi_T$ that integrates a differential equation over a finite time $T$. For example, if $x' = F(x,t)$, then $\Phi_T(x(0)) = x(T)$. If $x(T) = x(0)$, then $x(0)$ represents the initial data of a periodic orbit. We can assess the stability of this periodic orbit by computing the eigenvalues of the Poincaré map $\Phi_T$. Eigenvalues smaller than $1$ correspond to stable directions, and eigenvalues larger than $1$ correspond to unstable directions.
Now, suppose I have a partial differential equation. As a trivial example, take the wave equation $0=\ddot x - x''$, where $x = x(t,r)$ and dots refer to $t$ derivatives and primes refer to $r$ derivatives. I may still compute a map that evolves the system forward in $t$, however this "Poincaré map" is now a function of $r$.
Is there a way to interpret this new map in a similar way as the Poincaré map for ODEs?