Generalized Lyapunov exponent as a unified characterization of dynamical instabilities

Takuma Akimoto, Masaki Nakagawa, Soya Shinkai, Yoji Aizawa

    Research output: Contribution to journalArticlepeer-review

    6 Citations (Scopus)


    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.

    Original languageEnglish
    Article number012926
    JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
    Issue number1
    Publication statusPublished - 2015 Jan 30

    ASJC Scopus subject areas

    • Condensed Matter Physics
    • Statistical and Nonlinear Physics
    • Statistics and Probability


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