Prog. Theor. Phys. Vol. 63 No. 5 (1980) pp. 1804-1807

[ Full Text PDF : FREE ACCESS (360K) ]

Letters

Chaos, External Noise and Fredholm Theory

Yoshitsugu Oono and Yoichiro Takahashi^*

Research Institute of Industrial Science, Kyushu University, Fukuoka 812
^*Department of Mathematics, College of General Education, University of Tokyo, Komaba, Tokyo 153

(Received December 7, 1979)

Abstract:

The Fredholm theory of integral equations is applied to the Perron-Frobenius equation which determines the invariant measure of nonlinear difference equations. The resultant Fredholm determinant D is identical to 1/ζ. where ζ is the Artine-Mazure-Ruelle ζ-function. From the investigation of D we get the observable condition of the invariant measure of chaos, its stability against external noise, etc.

URL : http://ptp.ipap.jp/link?PTP/63/1804/
DOI : 10.1143/PTP.63.1804

References:

Y. Oono and M. Osikawa, Prog. Theor. Phys. 64 (1980), 54[PTP].
R. May, Nature 261 (1976), 459[CrossRef].
F. G. Tricomi, Intergral Equations (Interscience Publ. Inc., New York, 1957).
S. G. Mikhlin, Integral Equations (Pergamon Press, Oxford, 1964).
D. Ruelle, Thermodynamic Formalism (Encyclopedia of Mathematics and Its Applications vol. 5, Addison-Wesley, Reading, Mass, 1978).
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lect. Note Math. 470, Springer, New York, 1975).
Y. Oono, Prog. Theor. Phys. 59 (1978), 1029[PTP].
L. Block, Proc. AMS 72 (1978), 576.
A. Shibata, T. Mayuyama, M. Mizutani and N. Saito, preprint.
Sh. Ito, S. Tanaka and H. Nakada, Tokyo J. Math. 2 (1979), 221.
Sh. Ito and Y. Takahashi, J. Math. Soc. Jpn. 26 (1974), 33.
G. Wagner, Z. Wahrscheinlichkeiltstheorie 46 (1979), 317.
T. Kai and K. Tomita, preprint.
G. H. Hardy and J. E. Littlewood, Proc. London Math Soc. 22 (1924), 254.

Citing Article(s) :

Progress of Theoretical Physics Vol. 71 No. 4 (1984) pp. 851-854 :
Towards the Statistical Mechanics of Chaos

Yoichiro Takahashi and Yoshitsugu Oono
Progress of Theoretical Physics Supplement No.79 (1984) pp. 75-95 :
Coupling Sensitivity of Chaos

Hiroaki Daido
Progress of Theoretical Physics Supplement No.99 (1989) pp. 165-205 :
Large Deviation and Statistical Physics

Yoshitsugu Oono
Progress of Theoretical Physics Supplement No.99 (1989) pp. 263-287 :
Towards Thermodynamics of Spatiotemporal Chaos

Kunihiko Kaneko
Progress of Theoretical Physics Supplement No.130 (1998) pp. 29-44 :
Steady State Thermodynamics

Yoshitsugu Oono and Marco Paniconi