Quantitative universality for a class of weakly chaotic systems

Abstract

We consider a general class of intermittent maps designed to be weakly chaotic, i.e., for which the separation of trajectories of nearby initial conditions is weaker than exponential. We show that all its spatio and temporal properties, hitherto regarded independently in the literature, can be represented by a single characteristic function $\phi$. A universal criterion for the choice of $\phi$ is obtained within the Feigenbaum's renormalization-group approach. We find a general expression for the dispersion rate $\zeta(t)$ of initially nearby trajectories and we show that the instability scenario for weakly chaotic systems is more general than that originally proposed by Gaspard and Wang [Proc. Natl. Acad. Sci. USA {\bf 85}, 4591 (1988)]. We also consider a spatially extended version of such class of maps, which leads to anomalous diffusion, and we show that the mean squared displacement satisfies $\sigma^{2}(t)\sim\zeta(t)$. To illustrate our results, some examples are discussed in detail.