Remarkable similarities in distributions of dynamical observables in chaotic systems

TL;DR

This paper reveals that different dynamical observables in chaotic systems often share the same large deviation rate function, indicating a surprising universality in their statistical behavior.

Contribution

It introduces the concept of 'derived' functions to explain the observed similarity and shows that certain observables have N-independent, non-Gaussian distributions in the large-N limit.

Findings

Different observables share the same rate function despite different functions g(x)

Derived functions lead to N-independent, non-Gaussian distributions

Position observable and FTLE in certain maps are examples of derived functions

Abstract

The study of chaotic systems, where rare events play a pivotal role, is essential for understanding complex dynamics due to their sensitivity to initial conditions. Recently, tools from large deviation theory, typically applied in the context of stochastic processes, have been used in the study of chaotic systems. Here, we study dynamical observables, $A = \sum_{n=1}^N g(\textbf{x}_n)$, defined along a chaotic trajectory $\{\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_N\}$. For most choices of $g(\textbf{x})$, $A$ satisfies a central limit theorem: At large sequence size $N \gg 1$, typical fluctuations of $A$ follow a Gaussian distribution with a variance that scales linearly with $N$. Large deviations of $A$ are usually described by the large deviation principle, that is, $P(A) \sim e^{- N I(A/N)}$, where $I(a)$ is the rate function. We find that certain dynamical observables exhibit a remarkable statistical similarity: even when constructed with distinct functions $g_1(\textbf{x})$ and $g_2(\textbf{x})$, different observables are described by the same rate function. We provide a physical interpretation for this striking similarity by showing that $g_1(\textbf{x})-g_2(\textbf{x})$ belongs to a class of functions that we call ``derived''. Furthermore, we show that if $g(\textbf{x})$ itself is ``derived'', then the distribution of $A$ becomes independent of $N$ in the large-$N$ limit, and is generally non-Gaussian (although it is mirror-symmetric). We demonstrate that the position observable for certain open maps, used to model random walks and the finite-time Lyapunov exponent (FTLE) for the logistic map are of this derived form, thus providing a simple explanation for some existing results.