Critical parameters of the synchronisation's stability for coupled maps in regular graphs

TL;DR

This paper derives critical parameters for the stability of synchronization in coupled chaotic maps on regular and cyclic graphs, providing explicit formulas based on Laplacian eigenvalues and classifying graphs by topology.

Contribution

It introduces closed-form expressions for eigenvalues of regular graphs and extends stability analysis to cyclic graphs like k-cycles and k-M"obius ladders, highlighting topological effects.

Findings

Derived critical coupling strength, Lyapunov exponent, and link density for synchronization stability.

Provided explicit eigenvalue formulas for regular graphs and classified graphs by topology.

Compared stability differences between finite and infinite size graphs.

Abstract

Coupled Map Lattice (CML) models are particularly suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. Complete synchronisation in CMLs emerges when all maps have their state variables with equal magnitude, forming a spatially-uniform pattern that evolves in time. Here, we derive critical values for the parameters -- coupling strength, maximum Lyapunov exponent, and link density -- that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation symmetries). Our derivations are based on the Laplacian matrix eigenvalues, where we give closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of regular graphs and show that these graphs can be classified into two sets according to a topological condition (derived from the stability analysis). We also make derivations for two classes of cyclic graph: $k$-cycles (i.e., regular lattices of even degree $k$, which can be embedded in $T^k$ tori) and $k$-M\"obius ladders, which we introduce here to generalise the M\"obius ladder of degree $k = 3$. Our results highlight differences in the synchronisation manifold's stability of these graphs -- even for identical node degrees -- in the finite size and infinite size limit.