2025/05/14 by Lucianno Defaveri, Defaveri, Lucianno, Naftali R. Smith +1 · 1 citation
Economics, Econometrics and Finance · Physics and Astronomy · #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #Complex Systems and Time Series Analysis #FOS: Physical sciences #Quantum chaos and dynamical systems #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2505.09225
openalex publication_date 2025/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 = ∑n=1N g(xn), defined along a chaotic trajectory \x1, x2, …, xN\. For most choices of g(x), A satisfies a central limit theorem: At large sequence size N ≫ 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) ∼ 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 g1(x) and g2(x), different observables are described by the same rate function. We provide a physical interpretation for this striking similarity by showing that g1(x)-g2(x) belongs to a class of functions that we call ``derived''. Furthermore, we show that if g(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.