Linear encoding of the spatiotemporal cat

TL;DR

This paper introduces a linear encoding method for spatiotemporal chaos in a 1D lattice of coupled cat maps, enabling precise statistical predictions of pattern frequencies using a Green's function approach.

Contribution

It presents a novel linear symbolic encoding of spatiotemporal states and a systematic method to compute pattern measures in a chaotic lattice system.

Findings

Finite patterns can describe system states with exponential accuracy.

A Green's function approach effectively calculates pattern frequencies.

The method provides a new way to analyze spatiotemporal chaos statistically.

Abstract

The dynamics of an extended, spatiotemporally chaotic system might appear extremely complex. Nevertheless, the local dynamics, observed through a finite spatiotemporal window, can often be thought of as a visitation sequence of a finite repertoire of finite patterns. To make statistical predictions about the system, one needs to know how often a given pattern occurs. Here we address this fundamental question within a spatiotemporal cat, a 1-dimensional spatial lattice of coupled cat maps evolving in time. In spatiotemporal cat, any spatiotemporal state is labeled by a unique 2-dimensional lattice of symbols from a finite alphabet, with the lattice states and their symbolic representation related linearly (hence "linear encoding"). We show that the state of the system over a finite spatiotemporal domain can be described with exponentially increasing precision by a finite pattern of symbols, and we provide a systematic, lattice Green's function methodology to calculate the frequency (i.e., the measure) of such states.