Timing of Transients: Quantifying Reaching Times and Transient Behavior in Complex Systems

In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, Area under Distance Curve and Regularized Reaching Time, that capture two complementary aspects of transient dynamics. The first, Area under Distance Curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are "reluctant", i.e. stay distant from the attractor for long, or "eager" to approach it right away. Regularized Reaching Time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much "earlier" or "later" than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features.