Internal Time of a Dynamical System

Abstract

We have a dynamical system. The evolution of its state, X(t) at instant t, is given by a differential equation dX(t)/dt = f(X(t),t), independent of the environment. We propose to introduce a time s, or internal time, different from time t, or reference time. For this purpose we consider duration, or time elapsed between two instants. Reference duration, between instants t1 and t2 is obviously given by dr(t1,t2) = t2-t1. Any duration, for example internal duration di(t1,t2), must satisfy certain conditions. Once we have an internal duration di(t1,t2), we can generate an internal time s = di(t0,t). The choice of di(t1,t2) depends upon the “weight” of reference duration t2-t1, seen from the internal point of view, or equivalently that of infinitesimal reference duration dt between t and t+dt. We propose that the internal duration corresponding to reference duration dt is equal to (d(X(t)/dt)² dt. In a way (dX(t)/dt)² is an index of the “importance” of instant t. As an example, we consider an “explosive-implosive” dynamical system described by a certain evolution equation. The corresponding internal time varies from -∞ to +∞ while reference time varies from 0 to +∞. Interpretations (physiology, cosmology) are given.