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 $\varphi$. A universal criterion for the choice of $\varphi$ 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.

12 pages, updated version

References in corpus (8)

• Pesin-Type Identity for Weak Chaos
• Universality of algebraic laws in Hamiltonian systems
• Calculation of Superdiffusion for the Chirikov-Taylor Model
• Subexponential instability implies infinite invariant measure
• Chaotic Hamiltonian systems revisited: Survival probability
• Thermodynamic phase transitions for Pomeau-Manneville maps
• $1/f$ Spectrum and 1-Stable Law in One-Dimensional Intermittent Map with Uniform Invariant Measure and Nekhoroshev Stability
• On stable laws in a one-dimensional intermittent map equipped with a uniform invariant measure

Cited by in corpus (4)

• Slow Lévy flights
• Number of first-passage times as a measurement of information for weakly chaotic systems
• Exact invariant measures: How the strength of measure settles the intensity of chaos
• Scaling analysis of stationary probability distributions of random walks on one-dimensional lattices with aperiodic disorder