  Time-dependent potentials are common in galactic systems that undergo
significant evolution, interactions, or encounters with other galaxies, or when
there are dynamic processes like star formation and merging events. Recent
studies show that an ensemble approach along with the so-called snapshot
framework in dynamical system theory provide a powerful tool to analyze time
dependent dynamics.
  In this work, we aim to explore and quantify the phase space structure and
dynamical complexity in time-dependent galactic potentials consisting of
multiple components. We apply the classical method of Poincar\'e-surface of
section to analyze the phase space structure in a chaotic Hamiltonian system
subjected to parameter drift. This, however, makes sense only when the
evolution of a large ensemble of initial conditions is followed. Numerical
simulations explore the phase space structure of such ensembles while the
system undergoes a continuous parameter change. The pair-wise average distance
of ensemble members allows us to define a generalized Lyapunov-exponent, that
might also be time dependent, to describe the system stability. We revise the
system parameters for the Milky Way galaxy and provide a comprehensive
dynamical analysis of the system under circumstances where linear mass transfer
undergoes between the disk and bulge components of the model.
