Coupled Arnol'd cat maps on circulant graphs

TL;DR

This paper explores the chaotic dynamics of coupled Arnol'd cat maps on circulant graphs, analyzing their Lyapunov spectra, entropy, and period spectra, revealing that increased connectivity does not necessarily increase entropy.

Contribution

It introduces a symplectic coupling matrix for Arnol'd cat maps on circulant graphs and analyzes their chaotic properties, including entropy and spectral characteristics.

Findings

Entropy production does not increase with graph connectivity.

The system's Lyapunov spectra are analyzed in detail.

Spectra of periods on finite phase space are examined.

Abstract

This paper investigates the chaotic properties of Arnol'd cat maps (ACMs) coupled on the nodes of a circulant graph. By demanding that the system's evolution matrix be symplectic, we determine the coupling matrix, which is naturally interpreted as the adjacency matrix of a circulant graph. Specifically, the study analyses the system's Lyapunov spectra and Kolmogorov-Sinai (K-S) entropy. Numerical simulations yield the counterintuitive result that the entropy production does not increase as the connectivity of the graph increases, due to the translational symmetry of the circulant graph. Moreover, we analyse the spectra of the periods of the evolution matrix on a finite toroidal phase space of the dynamical system.