Statistics of Weakly Chaotic Systems

TL;DR

This paper reviews the theory of weakly chaotic systems, highlighting their significance, complexity, and recent advances, bridging stochastic and deterministic dynamics in understanding chaos.

Contribution

It provides a comprehensive overview of weakly chaotic systems, including new results, emphasizing their importance in modeling real-world processes and the challenges in their theoretical development.

Findings

Most chaotic systems are weakly chaotic.

Weakly chaotic systems are more complex to analyze than strongly chaotic ones.

Recent results advance understanding of weak chaos theory.

Abstract

One of the major breakthroughs in science of the last (20th) century was building a bridge between the worlds of stochastic (random) systems and deterministic (dynamical) systems. It was started by the celebrated 1958 paper by A.N.Kolmogorov \cite{Kolmo}, who called this new theory (and actually a new way of thinking about deterministic systems) stochasticity of dynamical systems. Later, this name was essentially replaced by a short (sexier but more vague ``chaos theory"). Kolmogorov's discovery demonstrated that the time evolution of deterministic systems could be indistinguishable from the evolution of purely random (stochastic) systems. Moreover, it has been later established that typical deterministic systems are chaotic. However, as well as stochastic systems, which could be more or less random (from random processes with independent values to random processes with long memory) , chaotic dynamical systems could be strongly or weakly chaotic. Actually, the majority of chaotic systems are weakly chaotic, especially those that are relevant models for various real-world processes. Naturally, the theory of weakly chaotic systems is less developed because it deals with more complicated problems than the studies of strongly chaotic systems. We present here a brief review of this theory, including recently published and some new results.