Large deviations in chaotic systems: exact results and dynamical phase transition

TL;DR

This paper analytically derives exact large deviation rate functions for several chaotic maps, uncovers a dynamical phase transition, and develops numerical tools for simulating rare events in non-invertible chaotic systems.

Contribution

It provides the first exact analytical rate functions for certain chaotic maps and identifies a dynamical phase transition, along with new numerical methods for rare event simulation.

Findings

Exact rate functions for doubling, tent, and logistic maps.

Evidence of a second order dynamical phase transition.

A new numerical tool for simulating atypical sequences in chaotic maps.

Abstract

Large deviations in chaotic dynamics have potentially significant and dramatic consequences. We study large deviations of series of finite lengths $N$ generated by chaotic maps. The distributions generally display an exponential decay with $N$, associated with large-deviation (rate) functions. We obtain the exact rate functions analytically for the doubling, tent, and logistic maps. For the latter two, the solution is given as a power series whose coefficients can be systematically calculated to any order. We also obtain the rate function for the cat map numerically, uncovering strong evidence for the existence of a remarkable singularity of it that we interpret as a second order dynamical phase transition. Furthermore, we develop a numerical tool for efficiently simulating atypical realizations of sequences if the chaotic map is not invertible, and we apply it to the tent and logistic maps.