Toshihiro Shimizu

Relaxation and bifurcation in Brownian motion driven by a chaotic force

Brownian motion driven by a chaotic sequence of iterates of a map F(y), which may depend on a bifurcation parameter, is discussed: ⋗n(t)=-γν(t)+ƒ(t), where ƒ(t)=Kyn+1 for nτ<t≤(n+1)τ (n=0, 1, 2,...) and Yn+1=F(Yn). The time evolution equation for the distribution function of the velocity is derived. If γτ⪢1, the distribution tends to the stationary one, which has a similar form as the invariant density of F(y). In the limit of τ→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.

Perturbation theory analysis of chaos

Abstract The theory developed in a previous paper (I) is applied to a time-dependent Hamiltonian system, which is governed by a three-dimensional differential system. The transition from quasiperiodicity to chaos is discussed theoretically by using the nonlinear scale method. The transition is compared with that in a dissipative system. The time evolution of the system is decomposed into a fastly varying part and a slowly varying part. The slowly varying part is shown to vacillate for some initial conditions. The vacillation is deterministic, but depends sensitively on the initial conditions. The theoretical result is shown to be in a good agreement with that of the numerical simulation.

Fractal structure and Gaussian distribution in chaotic Brownian motion

In Brownian motion driven by a chaotic sequence of iterates of a map F(y), x(t) = −γx(t) + f(t), where f(t) = yn + 1τ for nτ < t ≦ (n + 1)τ (n = 1, 2,…) and y n + 1 = F(yn), the fractal structure and the τ-dependence of the recurrence relation (xn + 1, xn), where xn = x (t = nτ), are studied. The recurrence relation is shown to change from a shape similar to F(y) into a diagonal strip, which does not depend on the details of F(y), if τ is decreased. Consequently it is shown that the stationary distribution of xn changes from a similar shape as the invariant density of F(y) into a Gaussian shape.

Chaotic force in Brownian motion

Brownian motion, driven by a chaotic force, is dicussed: v(t) = -γv(t) + ƒ(t). The force ƒ(t) takes the value y+ 1/√T for tn ⩽ t < tn + 1 (n = 0, 1, 2…,), where tn=Tσn-1j=0xj.

CHAOTIC BROWNIAN NETWORK

A system of many Brownian particles interacting via the chaotic force is proposed as a new model of neural network, which is called the chaotic Brownian network. The network is applied to the associative memory problem. The coupling constants are determined by the Hebb rule. It is shown that some neuron pairs, which are specified by the stored patterns, exhibit the synchronized motions with the same sign or with the opposite sign. The relation between the synchronized motion and the chaotic wandering motion is discussed. The network is shown to retrieve the stored patterns or the reverse ones systematically and very quickly. It is shown that the network can store and retrieve the number of patterns comparable to the number of neurons.

Thermal properties of low-resistive and semi-insulating GaAs wafers heated locally by Nd:Y3Al5O12 laser beam

Low‐resistive and semi‐insulating GaAs wafers were locally heated by a Nd:Y3Al5O12 laser beam (the wavelength: 1.06 μm). Remarkably different thermal properties were found between these two kinds of wafers. The reason should be ascribed to the function of the mid‐gap level (EL2) in GaAs wafers. The different thermal behaviors of the semi‐insulating GaAs from the semi‐insulating GaP, which has been reported previously, are also discussed.

Generalized Langevin equation with chaotic force

The generalized Langevin equation with chaotic force is investigated: x(t) = − ∫0tdt′φ(t,t′)x(t′) + ƒ(t), where φ(t,t′) = 《ƒ(t)ƒ(t′) 》《x2 》. The chaotic force ƒ(t) is defined by ƒ(t)=(yn+1 − 《y》τ 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.

Resonance phenomena in a harmonic oscillator driven by the chaotic force

The coherent nature of chaos is investigated in a simple harmonic oscillator driven by the chaotic force. The resonance phenomena between a harmonic oscillator and a chaotic oscillator are discussed in two cases: (a) the bifurcation parameter of the chaotic force is modulated by the amplitude of the harmonic oscillator and (b) the eigenfrequency of the harmonic oscillator is also modulated by the chaotic force in addition to (a). As a model of neural network, a system of many Brownian particles interacting via the chaotic force is proposed. The controlling of chaos is also discussed.

Stationary distribution of a nonlinear system driven by a chaotic force

A simple nonlinear system, driven by a chaotic force, is discussed: x(t) = [1 + ƒ(t)]x(t) − x(t)2. 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).

Perturbation theory analysis of chaos III. Period doubling bifurcations in a simple model

By using the nonlinear scale method the asymptotic solution of a three-dimensional differential system is calculated to elucidate the period-doubling bifurcation route to chaos. As an stochastic description of chaos a Langevin-type equation with a vacillation force is derived. The stochastic and deterministic nature of the vacillation force is studied.