Scaling properties of growing noninfinitesimal perturbations in space-time chaos

Abstract

We study the spatiotemporal dynamics of random spatially distributed noninfinitesimal perturbations in one-dimensional chaotic extended systems. We find that an initial perturbation of finite size $ε_0$ grows in time obeying the tangent space dynamic equations (Lyapunov vectors) up to a characteristic time $t_{\times}(ε_0) \sim b - (1/λ_{max}) \ln (ε_0)$, where $λ_{max}$ is the largest Lyapunov exponent and $b$ is a constant. For times $t < t_{\times}$ perturbations exhibit spatial correlations up to a typical distance $ξ\sim t^z$. For times larger than $t_{\times}$ finite perturbations are no longer described by tangent space equations, memory of spatial correlations is progressively destroyed and perturbations become spatiotemporal white noise. We are able to explain these results by mapping the problem to the Kardar-Parisi-Zhang universality class of surface growth.