title: Generalized Lyapunov exponent as a unified characterization of dynamical instabilities.
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.
authors: Department of Mechanical Engineering, Keio University, Yokohama 223-8522, Japan.
authors: Takuma
authors: T
authors: Akimoto
authors: Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan.
authors: Masaki
authors: M
authors: Nakagawa
authors: Research Center for the Mathematics on Chromatin Live Dynamics (RcMcD), Hiroshima University 739-8530, Japan.
authors: Soya
authors: S
authors: Shinkai
authors: Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan.
authors: Yoji
authors: Y
authors: Aizawa