Pausing for arbitrarily long times in dynamical systems

Abstract

It is well known that continuity in dynamical systems is not sufficient to guarantee uniqueness of solutions, but less obvious is that non-uniqueness can carry internal structure useful to characterize a system's dynamics. The non-uniqueness that concerns us here arises when an isolated non-differentiability of a flow results in spatial or temporal ambiguity of solutions. Spatial ambiguity can render a flow set valued after a specific event, and non-trivial examples are increasingly being seen in models of switching occurring in electronic or biological systems. Temporal ambiguity can mean that the same spatial trajectory may be traversed in different times, making an arbitrarily long pause at the non-differentiable point. We focus here on temporal indeterminacy and the extent to which it can be resolved. To investigate the typical forms, we take representative examples of the different conditions (non-differentiability, discontinuity or singularity) under which it occurs.