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Generalized Lyapunov exponent as a unified characterization of dynamical instabilities.

Authors
  • Akimoto, Takuma
  • Nakagawa, Masaki
  • Shinkai, Soya
  • Aizawa, Yoji
Type
Published Article
Journal
Physical Review E
Publisher
American Physical Society (APS)
Publication Date
Jan 01, 2015
Volume
91
Issue
1
Pages
12926–12926
Identifiers
PMID: 25679700
Source
Medline
License
Unknown

Abstract

The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent (exponential dynamical instability) is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of nonexponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., superexponential, exponential, and subexponential chaos. Using one-dimensional maps, we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the Lyapunov pair. In subexponential chaos, we show superweak 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.

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