Universal scaling of Lyapunov-exponent fluctuations in space-time chaos

Abstract

Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase-space. A recent numerical study of spatially-extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a non-trivial scaling behavior, D(L) ~ L^{-\gamma}, with the system size L. Here, we show that the wandering exponent \gamma can be expressed in terms of the roughening exponents associated with the corresponding "Lyapunov-surface". Our theoretical predictions are supported by the numerical analysis of several spatially-extended systems. In particular, we find that the wandering exponent of the first LE is universal: in view of the known relationship with the Kardar-Parisi-Zhang equation, \gamma can be expressed in terms of known critical exponents. Furthermore, our simulations reveal that the bulk of the spectrum exhibits a clearly different behavior and suggest that it belongs to a possibly unique universality class, which has, however, yet to be identified.