Deterministic nonlinear Brownian motion driven by a chaotic force
T. Shimizu and N. Morioka 
Graduate School of Engineering, and Department of Electrical
Engineering, Kokushikan University, Tokyo 154, Japan
To elucidate the bilateral aspect of chaos: the stochastic nature and the coherent nature, the following three deterministic Brownian motion models are proposed.
Model 1: 
.
Model 2: 
.
Model 3: 
.
In these models we study the Brownian motion with the chaotic force
f(t) instead of the usual random force. The chaotic force f(t)
changes chaotically at time intervals 
, 
for 
, where 
is the (n+1)th iterate
of a map F(y;r): 
. Here r
is the bifurcation parameter of the map and K is the magnitude of the
force. Since f(t) is deterministic, these models are not stochastic
but deterministic. Main results are summarized as follows.
Model 1
To study the characteristics of the system, we observe the system
stroboscopically at time intervals of and we get the recurrence
relation for 
.
(1) For large ( 
) the stationary distibution for
has the same form as that of the invariant density of F(y;r).
For small the stationary distribution is described as the
Gaussian form.
(2) If is decreased, the recurrence relation for exhibits
doubling and it has a fractal structure. Corresponding this change
the stationary distribution changes from the shape of the invariant
density to the Gaussian form keeping the fractal structure.
Model 2
The - and K-dependence of the stationary distribution is
discussed. It is shown that for small the stationary distribution
exhibits the drastic change according to K and the correlation of
. Chaos-induced-transition is discussed.
Model 3
In this model we study a simple harmonic oscillator coupled with a chaotic oscillator (a chaotic force). We will show that the harmonic oscillator is in a kind of resonance state with the chaotic oscillator through the bifurcation parameter modulation. Moreover, we will discuss the problem of controlling chaos.