Spatiotemporal structure of Lyapunov vectors in chaotic coupled-map lattices

TL;DR

This paper investigates the spatiotemporal behavior of Lyapunov vectors in chaotic coupled-map lattices, revealing localization, clustering, and scaling properties of characteristic vectors, contrasting with the delocalized backward vectors.

Contribution

It introduces a scale-invariant surface framework to analyze Lyapunov vectors, highlighting the distinct localization and scaling features of characteristic vectors in spatially extended chaos.

Findings

Characteristic Lyapunov vectors are spatially localized and cluster dynamically.

Backward Lyapunov vectors are delocalized and lack scaling properties.

Characteristic vectors exhibit dynamic scaling, unlike backward vectors.

Abstract

The spatiotemporal dynamics of Lyapunov vectors (LVs) in spatially extended chaotic systems is studied by means of coupled-map lattices. We determine intrinsic length scales and spatiotemporal correlations of LVs corresponding to the leading unstable directions by translating the problem to the language of scale-invariant growing surfaces. We find that the so-called 'characteristic' LVs exhibit spatial localization, strong clustering around given spatiotemporal loci, and remarkable dynamic scaling properties of the corresponding surfaces. In contrast, the commonly used backward LVs (obtained through Gram-Schmidt orthogonalization) spread all over the system and do not exhibit dynamic scaling due to artifacts in the dynamical correlations by construction.