Generalized Lyapunov exponent as a unified characterization of dynamical instabilities

TL;DR

This paper introduces the Lyapunov pair, based on the generalized Lyapunov exponent, to classify and quantify various types of dynamical instabilities, including super-exponential, exponential, and sub-exponential chaos, in one-dimensional maps.

Contribution

It proposes the Lyapunov pair as a unified measure for different chaos types and demonstrates super-exponential and sub-exponential chaos using one-dimensional maps.

Findings

Classified chaos into super-exponential, exponential, and sub-exponential types.

Quantified sub-exponential chaos, revealing super-weak chaos with slower-than-stretched exponential growth.

Analytically studied growth scaling using a continuous accumulation process related to infinite ergodic theory.

Abstract

The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of non-exponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., super-exponential, exponential, and sub-exponential dynamical instabilities. Using one-dimensional maps, we demonstrate super-exponential and sub-exponential chaos and quantify the dynamical instabilities by the Lyapunov pair. In sub-exponential chaos, we show super-weak chaos, which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth. The scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process, which is related to infinite ergodic theory.