On the Weibull distribution appeared in chaotic systems

Yoji Aizawa

Department of Applied Physics, Faculty of Science and Engineering,
Advanced Institute for Complex Systems, Waseda University, Tokyo, Japan

Various kinds of scaling laws are often observed in complex dynamical systems; for instance, SOC and punctuated equilibrium near the edge of chaos, 1/f spectrum, long time tails and anomalous diffusion in non-hyperbolic systems, where the self-similarity structures play essential roles in dynamical space and induce the breaking of central limit theorem for gaussian regime.

In the present paper we discuss three complex dynamical systems with non-gaussian scaling regime described by the Weibull distribution. Though the universality of Weibull distribution functions has not yet been made clear, but it is surmised that the Weibull regime is omnipresent in the systems under consideration.

1. Hamiltonian systems

Weibull distributions in hamiltonian dynamics were studied in (1): Mixmaster universe model [Prog. Theor. Phys. 98 No.6 (1997), 1225], and in (2) Cluster formation [Prog. Theor. Phys. 103 (2000), 519 ; Suppl. No.139 (2000), 1]. In particular, it was shown that the Arnold diffusion can be explained in terms of the universality of Log-Weibull distributions.

Here we discuss a new simplified model of random potential scattering, where Weibull distributions are numerically obtained.

2. Time dependent non-stationary chaotic map

It is known that the modified Bernoulli map reveals strong intermittency and anomalous large deviation properties [Prog. Theor. Phys. 99 (1989), 149 and 90 No.3 (1993), 547].

Here we consider the time dependent process where the system parameter is varying in the course of time. Surprising results are the followings ; the time dependent intermittency obeys the Weibull distribution in the intermediate long time scale, but in the intrinsic long time scale it obeys the Log-Weibull distribution.

3. A model of financial market

It is known in econophysics studies that the distribution of returns (change of price) obeys the stable ($L\acute{e}vy$) distribution, but many theoretical models for financial market do not always display the stable law.

Here we study a multi-agent model [Chaos, Solitons & Fractals 11 (2000), 1077 ; 1739, Physica A 287 (2000), 507], and point out that the return distributions are well adjusted by the Weibull distribution function in wide parameter range.