The Nature of Chaos in a Simple Dynamical System

1979 ◽  
Vol 34 (11) ◽  
pp. 1283-1289 ◽  
Author(s):  
Akira Shibata ◽  
Toshihiro Mayuyama ◽  
Masahiro Mizutani ◽  
Nobuhiko Saitô
Keyword(s):  
Dynamical System ◽  
Distribution Function ◽  
Probability Distribution ◽  
Correlation Functions ◽  
Probability Distribution Function ◽  
Time Correlation ◽  
Time Correlation Functions ◽  
One Dimensional ◽  
Initial Distributions ◽  
Simple Dynamical System

A simple one-dimensional transformation xn = axn-1 + 2 - a (0 ≦ xn-1 ≦ 1 - 1 / a ) , xn = a( 1 - xn-1) (1 - 1 a ≦ xn-1 ≦ 1) (1 ≦ a ≦ 2) is investigated by introducing the probability distribution function Wn(x). Wn ( x ) converges when n → oo for a > V 2 , but oscillates for 1 < a ≦ V2. The final distribution of Wn(x) does not depend on the initial distributions for a > V2, but does for 1 < a ≦V2 Time-correlation functions are also calculated

Theoretical Study of Time Correlation Functions in a Discrete Chaotic Process

1978 ◽  
Vol 33 (12) ◽  
pp. 1455-1460 ◽  
Author(s):  
Hirokazu Fujisaka ◽  
Tomoji Yamada
Keyword(s):  
Correlation Function ◽  
Correlation Functions ◽  
Characteristic Time ◽  
Theoretical Study ◽  
Time Correlation ◽  
Time Correlation Function ◽  
Point Of View ◽  
Time Correlation Functions ◽  
Chaotic Motions ◽  
One Dimensional

Abstract One-dimensional discrete chaotic processes are studied from a statistical-dynamical point of view. A set of equations which describe the behavior of the time correlation functions is derived with the aid of Mori’s projector formalism. A condition under which a process is Markoffian is obtained, and an approximate method is developed for a non-Markoffian process. As an illustration, a time correlation function for a simple system is calculated and the comparison with results of computer simulations is made. The relation between the instability of a trajectory and the characteristic time of chaotic motions is also discussed.

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