Exponential localization of singular vectors in spatiotemporal chaos

TL;DR

This paper demonstrates that in spatiotemporal chaotic systems, singular vectors localize exponentially in space, following a universal power law, with implications for improving forecasting methods.

Contribution

It reveals the exponential spatial localization of singular vectors and links their behavior to the KPZ equation, introducing a universal scaling law for their localization.

Findings

Singular vectors exponentially localize in space in spatiotemporal chaos.

A universal power law $ au^{- ext{gamma}}$ describes the localization.

The same exponent $ ext{gamma}$ explains deviations in Lyapunov exponents.

Abstract

In a dynamical system the singular vector (SV) indicates which perturbation will exhibit maximal growth after a time interval $\tau$. We show that in systems with spatiotemporal chaos the SV exponentially localizes in space. Under a suitable transformation, the SV can be described in terms of the Kardar-Parisi-Zhang equation with periodic noise. A scaling argument allows us to deduce a universal power law $\tau^{-\gamma}$ for the localization of the SV. Moreover the same exponent $\gamma$ characterizes the finite-$\tau$ deviation of the Lyapunov exponent in excellent agreement with simulations. Our results may help improving existing forecasting techniques.