  The recent discovery of a spiral pattern in the vertical kinematic structure
in the solar neighborhood provides a prime opportunity to study non-equilibrium
dynamics in the Milky Way from local stellar kinematics. Furthermore, results
from simulations indicate that even in a limited volume, differences in stellar
orbital histories allow us to trace variations in the initial perturbation
across large regions of the disk. We present $\texttt{ESCARGOT}$, a novel
algorithm for studying these variations in both simulated and observed data
sets. $\texttt{ESCARGOT}$ automatically extracts key quantities from the
structure of a given phase spiral, including the time since perturbation and
the perturbation mode. We test $\texttt{ESCARGOT}$ on simulated data and show
that it is capable of accurately recovering information about the time since
the perturbation occurred as well as subtle differences in phase spiral
morphology due to stellar location in the disk at the time of perturbation. We
apply $\texttt{ESCARGOT}$ to kinematic data from data release 3 of the ${\it
Gaia}$ mission in bins of guiding radius. We show that similar structural
differences in morphology occur in the ${\it Gaia}$ phase spirals as a function
of stellar orbital history. These results indicate that the phase spirals are
the product of a complex dynamical response in the disk with large-scale
coupling between different regions of phase space.
