A Quantification of Long Transient Dynamics
Differential equations , Transient dynamics , Transient centers , Transient centres
We present a systematic study of transient dynamics starting with a technical definition of transient points which are initial data of an autonomous system of ordinary differential equations that can lead to “long transient dynamics.” We then define transient centers, which are points in the state space that cause long transient behaviors, and reachable transient centers, which are transient centers that can be reached from initializations that do not need to be nearby. These points give rise to dynamics where a prescribed observable changes arbitrarily slowly for arbitrarily long time durations. We demonstrate the many interesting properties of transient centers, including how it easily translates from point to point: if an initial point is a transient center then so are all the points in its entire trajectory forward and backward in time, as well as any limit points. Finally, we also explore how these ideas are related to established concepts in dynamical systems such as slow-fast systems, turning points and Lyapunov regularity.