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

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.