Comparing chaotic and random dynamical systems

Abstract

This is a slightly expanded version of the plenary lecture the author gave at the International Congress on Mathematical Physics 2018 in Montreal, Canada. Reported are some work of the author and collaborators as well as related results of others on two kinds of dynamical systems: the first kind is deterministic (in the sense that nothing is left to chance) but is chaotic and unpredictable, and the second kind has a stochastic component in addition to a purely deterministic one. These two kinds of dynamical systems are compared and contrasted. The main points are that time series of observations from chaotic deterministic systems obey some of the same probabilistic limit laws as genuinely random stochastic processes, but random dynamical systems enjoy nicer properties and are technically more tractable.