Generalized circulant matrix in Coupled Map Lattices

TL;DR

This paper demonstrates that a generalized circulant matrix fundamentally underpins weakly Coupled Map Lattices, enabling explicit analytical solutions for their temporal evolution through its inverse, regardless of the coupling form.

Contribution

It introduces the use of a generalized circulant matrix as a core analytical tool for weakly CMLs, providing explicit inverse calculations for the first time.

Findings

Explicit inverse matrix derived for arbitrary order

Analytical solutions obtained using first-order approximation

Applicable to any coupling form in weakly CMLs

Abstract

In this paper it is shown that a generalized circulant matrix underlies every weakly Coupled Map Lattice (CML), independently of the form of the coupling term. Therefore, this matrix will appear always perturbative methods are used to get the analytical solutions. In fact, the inverse of this matrix provides the analytical solution of the CML after using first order approximation methods. This inverse matrix, of arbitrary order, is explicitly calculated, thus providing the analytical expression for the temporal evolution of the CML.