Stationary distribution of a nonlinear system driven by a chaotic force

A simple nonlinear system, driven by a chaotic force, is discussed: . 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).