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Stationary distribution of a nonlinear system driven by a chaotic force

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  • Shimizu, T.
  • Morioka, N.

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).

Suggested Citation

  • Shimizu, T. & Morioka, N., 1995. "Stationary distribution of a nonlinear system driven by a chaotic force," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 218(3), pages 390-402.
    • Handle: RePEc:eee:phsmap:v:218:y:1995:i:3:p:390-402
    DOI: 10.1016/0378-4371(95)00149-2
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    1. Talaat Abdel-Malek, 1976. "Some Aspects of Exchange Risk Policies under Floating Rates," Journal of International Business Studies, Palgrave Macmillan;Academy of International Business, vol. 7(2), pages 89-98, June.
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    Cited by:

    1. Shimizu, Toshihiro, 1996. "Resonance phenomena in a harmonic oscillator driven by the chaotic force," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 234(1), pages 371-385.

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