Exploring the role of diffusive coupling in spatiotemporal chaos

TL;DR

This paper investigates how diffusive coupling influences spatiotemporal chaos in a lattice of coupled maps, revealing how diffusion strength affects Lyapunov exponents, spatial structures, and mode localization.

Contribution

It provides an analytical and numerical analysis of the impact of diffusive coupling on chaos, including Lyapunov spectra, fractal dimensions, and CLV spatial features.

Findings

Lyapunov exponent decreases with increasing diffusion, then stabilizes.

Chaotic dynamics are composed of physical modes across conditions.

Leading CLV becomes less localized as diffusion strength increases.

Abstract

We explore the chaotic dynamics of a large one-dimensional lattice of coupled maps with diffusive coupling of varying strength using the covariant Lyapunov vectors (CLVs). Using a lattice of diffusively coupled quadratic maps we quantify the growth of spatial structures in the chaotic dynamics as the strength of diffusion is increased. When the diffusion strength is increased from zero, we find that the leading Lyapunov exponent decreases rapidly from a positive value to zero to yield a small window of periodic dynamics which is then followed by chaotic dynamics. For values of the diffusion strength beyond the window of periodic dynamics, the leading Lyapunov exponent does not vary significantly with the strength of diffusion with the exception of a small variation for the largest diffusion strengths we explore. The Lyapunov spectrum and fractal dimension are described analytically as a function of the diffusion strength using the eigenvalues of the coupling operator. The spatial features of the CLVs are quantified and compared with the eigenvectors of the coupling operator. The chaotic dynamics are composed entirely of physical modes for all of the conditions we explore. The leading CLV is highly localized and the localization decreases with increasing strength of the spatial coupling. The violation of the dominance of Oseledets splitting indicates that the entanglement of pairs of CLVs becomes more significant between neighboring CLVs as the strength of the diffusion is increased.