Convective Lyapunov Spectra

TL;DR

This paper extends the concept of convective Lyapunov exponents to a full spectrum, revealing a critical density where the spectrum's behavior changes discontinuously, linked to a change in the spectrum's concavity.

Contribution

It introduces a generalized convective Lyapunov spectrum and identifies a critical density causing a discontinuous change in the spectrum's velocity dependence.

Findings

Existence of a critical density $n_c$ with spectrum discontinuity.

Discontinuous dependence of the spectrum on velocity beyond $n_c$.

Change of concavity in the temporal Lyapunov spectrum at $n_c$.

Abstract

We generalize the concept of convective (or velocity-dependent) Lyapunov exponent $\Lambda(v)$ to an entire spectrum $\Lambda(v,n)$. Our results are supported by the consistency between the outcome of the chronotopic approach [{\it S. Lepri et al. J. Stat. Phys., 82 5/6 (1996) 1429}] and a more direct method. There exists a critical integrated density $n=n_c$, beyond which the convective exponent exhibits a discontinuous dependence on the velocity, which originates from the appearance of multiple branches. This phenomenon can be traced back to a change of concavity of the so-called {\it temporal} Lyapunov spectrum for $n>n_c$, which is therefore a dynamical invariant.