Ryuji Ishizaki

Randomization and Memory Functions of Chaos and Turbulence

The chaotic orbits of dynamical systems become random and stochastic on long timescales due to the orbital instability of chaos. This randomization of chaotic orbits gives rise to the dissipation of the macroscopic kinetic energy supplied by an external force into the random kinetic energy of chaos or turbulence and leads to various transport processes. The randomization of chaotic orbits is formulated in terms of a memory function which describes the loss of memory of the initial states due to the orbital instability. Then the nonlinear term of the evolution equation that causes chaos or turbulence is transformed into the sum of a fluctuating force and a memory-function term. Thus the deterministic evolution equation is found to become a Markovian stochastic equation on long timescales. Then, considering the chaos-induced friction of a forced pendulum and the molecular and turbulent viscosity of incompressible fluids, we explore the memory functions and the fluctuationdissipation formula for transport coefficients in order to establish a statistical-mechanical approach to chaos and turbulence. There are two kinds of time-correlation functions defined by the long time-average over

Characterization of a system described by Kuramoto–Sivashinsky equation with Lyapunov exponent

The characteristics of a system described by Kuramoto–Sivashinsky equation are obtained through the statistics of a mean Lyapunov exponent. This mean Lyapunov exponent takes large values and fluctuates large when the system is disordered temporally and spatially. This behavior of the spatially extended system is captured clearly by the probability distribution function for the time averaged one of the mean Lyapunov exponent.

Memory Function Approach to Chaos and Turbulence and the Continued Fraction Expansion

The chaotic orbits of dynamical systems are deterministic and predictable on short timescales τ γ , but they become stochastic and random on long timescales T M (» τ γ ) due to the orbital instability of chaos. This randomization of chaotic orbits has been formulated recently by deriving a non-Markovian stochastic equation for macrovariables in terms of a fluctuating force and a memory function. In order to develop this memory function approach to chaos and turbulence, we explore the following problems by studying the Duffing oscillator and the Navier-Stokes equation for an incompressible fluid: 1) the physical meaning of the projection of macrovariables A(t) onto A(0); 2) the method of calculating the short-lived motion with short timescale τ γ , which determines the memory functions and the macroscopic transport coefficients due to chaos and turbulence; 3) the continued fraction expansion of the memory function, and the order estimation of short timescales τ γ and long timescales τM; 4) the relation between the memory function and the time correlation function of a nonlinear force, which gives computable theoretical expressions for the macroscopic transport coefficients.

Time Correlations and Diffusion of a Conservative Forced Pendulum

The time correlations and diffusion of chaotic orbits in a periodically forced pendulum without friction are studied. The pendulum exhibits a Poincare section with period T at times t = jT (j =0 , 1, 2, ··· ). The time-correlation function C(t) ≡� p(t)p(0)� of the angular velocity p(t) oscillates with period T ,e ven ast →∞ , since the average quantities of the system have a periodicity with period T , due to the periodic external force. Studying the approach to asymptotic oscillation, we find that the time-correlation function C(t) exhibits an inverse power decay t −(β−1) (1 <β <2), where there exist islands of accelerator-mode tori. Then, it is also shown that the power spectrum Ip(ω )o fp(t) obeys an inverse power law ω −(2−β) for small frequency ω � 2π/T. We also calculate the mean square displacement σ 2 (n) of the angular variable qn ≡ q(nT ) on the Poincare section, and show that σ 2 (n) ∝ n ζ (ζ =3 − β) for n →∞ , leading to anomalous diffusion with 1 <ζ< 2.

The Memory Function and Chaos-Induced Friction in the Chaotic Hénon-Heiles System

A non-Markovian linear stochastic equation for the momentum p y (t) is derived for the purpose of clarifying transport processes in the chaotic Henon-Heiles system with the aid of the Mori projection operator formalism. For the time correlation function C y (t) = of the coordinate y(t), this leads to an integrable linear evolution equation. Then, the memory function γ(t) enables us to define a frequency-dependent chaos-induced friction coefficient of the system, γ(iω). We show that this friction coefficient is related to the time correlation function Φ(t) of a nonlinear force f(t), which can be computed numerically. Thus, in the case that the total energy is E = 1/6, it turns out that the structure of the frequency-dependent friction coefficient γ r (ω) consists of three sharp peaks at frequencies ω = 0, 0.859 and 1.891. This leads to a three-term approximation of the memory function, γ(t), with a correlation time τ r ∼ 5T (with T = 2π). It is also shown that the structure of the power spectrum I y (ω) of y(t) consists of four sharp peaks at frequencies ω = 0, 0.500, 0.797 and 1.000. This leads to a four-term approximation of the time correlation function C y (t) with a correlation time τ M ∼ 6T. The frequencies and line widths of the sharp peaks of I y (ω) are given by the friction coefficient γ(iω).

Application of Tsallis Nonextensive Statistics to the Anomalous Diffusion of the Standard Map

The anomalous diffusion due to accelerator-mode islands in the Standard Map is analyzed with the aid of Tsallis nonextensive statistics. In this treatment, we introduce a new variable x, which represents the displacement per jump while the chaotic orbit is trapped by the accelerator-mode islands. We have shown numerically that the one-jump distribution function p(x) is qualitatively similar to the function pq(x) derived using the maximum Tsallis entropy principle with appropriate conditions. [C. Tsallis, J. Stat. Phys. 52 (1988), 479; Phys. Lett. A 195 (1994), 329.] We find that the n-jump distribution function p(x, n) converges to the n-jump distribution function pq(x, n )= 1 n1/γ pq( x n1/γ ) obtained from the

Pathological Anomalous Diffusion Generated by a Generalized Shift Map.

The dynamics of a decimal point of two-sided infinite sequences (... a t (-2) a t (-1) . a t (0) a t (1) ...) is studied, where the motion of the point is governed by a generalized shift (GS) map. ...

Memory Spectra and Lorentzian Power Spectra of the Chaotic Duffing Oscillator

γr(ω )a nd ˆ γi(ω), respectively, we find that the power spectrum Iˆ(ω) of the momentum ˆ p(t )c an be written in terms of ˆ Ω0 ,ˆ γr(ω) andi(ω). For a Duffing oscillator with molecular viscosity γ 0 =0 .5 and an external force with amplitude b =0 .55 and frequency ω0 =1 .2, we find thatr(ω) has one sharp peak at frequency ω =1 .80 and a few small peaks. It is also shown that the power spectrum Iˆ p(ω) has two sharp peaks at frequencies ω1 =0 .509 and ω2 =1 .89 and one line spectrum at ω = ω0, leading to a two-peaks approximation to the time correlation function Cp(t )w ith correlation time τM ∼ 4.69T ,( T =2 π/ω0). Then, it is shown that the structure of the ω1-peak can be represented by an asymmetric Lorentzian peak.

Anomalous Diffusion Induced by Random Walks with Hierarchical Long-Range Memory

A discrete-time dual random walk is presented whose random increment v n consists of the sum of the first stage random increments u i as v n ≡|∑ i =1 n -1 u i | γ u n . Our numerical study shows that its exponent ζ of the mean square displacement linearly depends on the parameter γ as ζ= 1+γ. Critical phenomena is observed when u i is generated by iterated maps where the critical exponent α of the diffusion coefficient also linearly depends on the parameter γ as α=α 0 + c γ.

Non-Stationary Anomalous Diffusion Produced by a Generalized Shift Map

We study random walks of a decimal point of a two-sided infinite sequence (... a t (-2) a t (-1) . a t (0) a t (1) ...) where the point is driven by a generalized shift (GS) map, and find the super-diffusion whose distribution does not obey a Levy distribution. The diffusion depends on the rule of GS map, and we calculate the mean square displacement σ 2 ( n ) ∝ n ζ for diffusive dynamics rules and a fluctuated drift rule where a parameter P is introduced which represents the probability of a ( i +1) 0 whose value is the same with a ( i ) 0 . We find that ζ depends on P for the diffusive rules, whereas ζ is independent of P for the fluctuated drift rule.