Relaxation and bifurcation in Brownian motion driven by a chaotic force

✍ Scribed by Toshihiro Shimizu

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 796 KB
• Volume: 164
• Category: Article
• ISSN: 0378-4371
• DOI: 10.1016/0378-4371(91)90106-m

✦ Synopsis

Brownian motion driven by a chaotic sequence of iterates of a map F(y), which may depend on a bifurcation parameter, is discussed: 6(t)= -yv(t)+ f(t), where f(t)= Kyn, ~ for nr < t <~ (n + 1)r (n = 0, 1,2 .... ) and y,,_~ = F(y,,). The time evolution equation for the distribution function of the velocity is derived. If 3'~>> 1, the distribution tends to the stationary one, which has a similar form as the invariant density of F(y). In the limit of r--,0 the distribution function satisfies a Fokker-Planck type equation with memory effects. The dependence of relaxation processes on the bifurcation parameter is also investigated. The fluctuation-dissipation theorem is discussed. The theoretical results are shown to be in a good agreement with numerical ones, which have been done for the tent map and the logistic map.