Abstract
The Brownian motion driven by the chaotic force, which changes chaotically at time interval τ and has the deterministic nature, is discussed in comparison with the conventional Brownian motion, which is driven by the stochastic random noise. The chaotic sequence is generated by some mapping function. In the case of a linear Langevin equation it is shown that for large τ the stationary distribution has the shape of the invariant density of the mapping function and for small τ the stationary distribution can be described by the Gaussian form, which does not depend on the detail of the mapping function. The last case corresponds to the conventional Brownian motion with the stochastic random noise. The above two characteristic stationary distributions are shown to be connected via the fractal structure, if τ is decreased. In the case of a nonlinear Langevin equation the relation between the “chaotic integral” and the stochastic integral is discussed. For small τ the Fokker-Planck equation associated with the nonlinear deterministic Langevin equation is shown to coincide with that of Storatonovich type.
