  When quantifying the time spent in the transient of a complex dynamical
system, the fundamental problem is that for a large class of systems the actual
time for reaching an attractor is infinite. Common methods for dealing with
this problem usually introduce three additional problems: non-invariance,
physical interpretation, and discontinuities, calling for carefully designed
methods for quantifying transients.
  In this article, we discuss how the aforementioned problems emerge and
propose two novel metrics, Regularized Reaching Time ($T_{RR}$) and Area under
Distance Curve (AUDIC), to solve them, capturing two complementary aspects of
the transient dynamics.
  $T_{RR}$ quantifies the additional time (positive or negative) that a
trajectory starting at a chosen initial condition needs to reach the attractor
after a reference trajectory has already arrived there. A positive or negative
value means that it arrives by this much earlier or later than the reference.
Because $T_{RR}$ is an analysis of return times after shocks, it is a
systematic approach to the concept of critical slowing down [1]; hence it is
naturally an early-warning signal [2] for bifurcations when central statistics
over distributions of initial conditions are used.
  AUDIC is the distance of the trajectory to the attractor integrated over
time. Complementary to $T_{RR}$, it measures which trajectories are reluctant,
i.e. stay away from the attractor for long, or eager to approach it right away.
  (... shortened for arxiv listing, full abstract in paper ...) New features in
these models can be uncovered, including the surprising regularity of the
Roessler system's basin of attraction even in the regime of a chaotic
attractor. Additionally, we demonstrate the critical slowing down
interpretation by presenting the metrics' sensitivity to prebifurcational
change and thus how they act as early-warning signals.
