Generalized Langevin equation with chaotic force

The generalized Langevin equation with chaotic force is investigated: , where φ(t,t′) = ƒ(t)ƒ(t′) / x2 . The chaotic force ƒ(t) is defined by for nτ < t ≤ (n + 1)τ (n= 0,1,2,…), where yn+1 is a chaotic sequence: yn+1 = F(yn). The time evolution of x(t), which is generated by the chaotic force, is discussed. The approach of the distribution function of x to a stationary distribution is studied. It is shown that the distribution function satisfies the Fokker-Planck type equation with the memory effect in the small τ limit. The relation between the invariant density of F (y) and the stationary distribution of x is discussed