2014/12/22 by Takuma Akimoto, Masaki Nakagawa, Soya Shinkai +1 · 1 citation
Mathematics · Physics and Astronomy · #math.DS #nlin.CD
paper · pdf · doi:10.1103/physreve.91.012926
8 pages, 4 figures
arxiv created 2014/12/22 · arxiv updated 2015/06/23
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.