Sharkovskii’s theorem and the limits of digital computers for the simulation of chaotic dynamical systems

Abstract

Chaos is a unique paradigm in classical physics within which systems exhibit extreme sensitivity to the initial conditions. Thus, they need to be handled using probabilistic methods commonly based on ensembles. However, initial conditions generated by digital computers fall within the sparse set of discrete IEEE floating point numbers which have non-uniform distributions along the real axis. Therefore, there are many missing initial conditions whose absence might be expected to degrade the computed statistical properties of chaotic systems. The universality of this problem is enshrined in Sharkovskii’s theorem which is the simplest mathematical statement of the fact that no finite number representation of a chaotic dynamical system can account for all of its properties and shows that the precision of the representation limits the accuracy of the resulting digital behaviour.