Stationary distribution of a nonlinear system driven by a chaotic force

T. Shimizu and N. Morioka

Physica A: Statistical Mechanics and its Applications, 1995, vol. 218, issue 3, 390-402

Abstract: A simple nonlinear system, driven by a chaotic force, is discussed: ẋ(t) = [1 + ƒ(t)]x(t) − x(t)2. The chaotic force ƒ(t) is defined by ƒ(t) = Kg(Yn+1)/√τ for nτ < t ≤ (n + 1)τ, n = 0, 1, 2, …, where Yn+1 is a chaotic sequence of a map F(y): Yn+1, −0.5 ≤ Yn ≤ 0.5. As g(y) two cases are considered: (a) g(Yn+1) = Yn+1 − 〈Y0〉 and (b) g(Yn+1) = Yn+1/∥Yn+1∥ The relaxation process of this system is investigated theoretically. The τ- and K-dependence of the stationary distribution of x is discussed. It is shown that for small τ the stationary distribution exhibits a drastic change according to K and the correlation of Yn. The fractal structure of the stationary distribution is found. The theoretical results are shown to be in a good agreement with numerical ones, which have been done for the logistic map as F(y).

Date: 1995
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:218:y:1995:i:3:p:390-402

DOI: 10.1016/0378-4371(95)00149-2

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