Quantitative universality for a class of weakly chaotic systems

TL;DR

This paper introduces a universal framework for weakly chaotic systems using a characteristic function, providing a unified description of their dynamics, instability, and anomalous diffusion, with broad applicability and new insights into their behavior.

Contribution

The paper develops a universal criterion for weakly chaotic maps via a characteristic function, unifying their spatio-temporal properties and extending the instability scenario.

Findings

A single characteristic function $oldsymbol{}$ describes all properties of weakly chaotic maps.

Derived a general expression for the dispersion rate $oldsymbol{(t)}$ of nearby trajectories.

Showed that weak chaos leads to anomalous diffusion with mean squared displacement proportional to $oldsymbol{(t)}$.

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.