Prog. Theor. Phys. Vol. 69 No. 5 (1983) pp. 1416-1426

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Generalization of Baker's Transformation

— Chaos and Stochastic Process on a Smale's Horse-Shoe —

Yoji Aizawa and Chikara Murakami

Department of Physics, University of Kyoto, Kyoto 606

(Received December 16, 1982)

Abstract:

The area-preserving baker's transformation is extended in order to include the area non-preserving one, which is the model for a simple horse-shoe structure. The random motion on the horse-shoe is studied precisely, and the invariant density and the correlation function are obtained analytically and numerically.
The random walk on the horse-shoe structure is transformed into the generalized Brownian motion with the attractor, and the non-analytic density of the generalized baker's transformation is approximated by an analytic function. The topological similarity between the Lorenz chaos and the generalized baker's transformation is briefly discussed.

URL : http://ptp.ipap.jp/link?PTP/69/1416/
DOI : 10.1143/PTP.69.1416

References:

S. Smale, Bull. Amer. Math. Soc. 73 (1967), 747.
Y. Aizawa, Prog. Theor. Phys. 68 (1982), 64[PTP].
W. Feller, `An Introduction to Probability Theory and Its Applications', vol. II (John Wiley & Sons, Inc., 1966).
N. Saito, J. Phys. Soc. Jpn. 51 (1982), 374[JPSJ].
K. Kitahara, W. Horsthemke, R. Lefever and Y. Inaba, Prog. Theor. Phys. 64 (1980), 1233[PTP].
T. T. Soong, `Random Differential Equations in Science and Engineering' (Acad. Press, N. Y., 1973).
E. N. Lorenz, J. Atoms. Sci. 20 (1963), 130.
Y. Aizawa and T. Uezu, Prog. Theor. Phys. 67 (1982), 982[PTP].

Citing Article(s) :

Progress of Theoretical Physics Supplement No.79 (1984) pp. 96-124 :
Statistical Mechanics of Intermittent Chaos

Yoji Aizawa, Chikara Murakami and Tamotsu Kohyama