Syuji Miyazaki

On-off intermittency in oscillatory media

Abstract On-off intermittency observed in three types of spatially extended dynamical systems is reported. This is done by examining the linear stability of spatially synchronized state under a spatially inhomogeneous fluctuation. When the system size slightly exceeds the critical size and a single inhomogeneous mode becomes unstable, systems exhibit typical on-off intermittency, whose statistical characteristics such as distribution of burst amplitude, Fourier spectrum and distribution of laminar duration turn out to be in agreement with those known in the on-off intermittency for small degrees of freedom system.

Crossover between Anomalous Superdiffusion and Normal Diffusion in Oscillating Convection Flows

Anomalous diffusion found in fluid systems is studied. Diffusion constants and mean square displacements are analytically obtained on the basis of the continuous-time random walk (CTRW) velocity model, and the values are compared with those obtained from model simulations employing dissipative dynamics describing oscillating convection flows. Good agreement is obtained.

On-Off Intermittency in a Four-Dimensional Poincaré Map

A four-dimensional Poincare map is obtained from a coupling between a pair of nonlinear oscillators and it shows on-off intermittency under proper conditions. The power spectrum, and the distribution of on-off variable and that of laminar duration are numerically obtained and compared with analytical results of the multiplicative noise model, which is a reduced form of the large friction limit of the original Poincare map. The coupled system considered here is a good candidate with which to observe on-off intermittency experimentally.

Global Bifurcations and Fluctuation Spectra of Local Expansion Rates in Nonlinear Dynamical Systems

The definition of the fluctuation spectrum of local expansion rates of nearby orbits is extended to a global region of phase space. It contains information on repellers as well as attractors coexisting in the region. Lyapunov exponents of attractors and repellers, the escape rates for repellers and some structures of these invariant sets are represented by this spectrum. Creations and annihilations of these invariant sets, the change of their nature and interrelations, occurring through global bifurcations when a control parameter value is varied, can be seen clearly in terms of this spectrum. On the basis of the statistical· mechanical formalism for this spectrum, some of the invariant sets coexisting in phase space for a fixed value of control parameter are interpreted as in phase equilibrium under the weighted average. At the bifurcation points the present first order phase transitions coincide with those for attractors found in previous papers. Long-term behaviors in nonlinear dynamical systems are understood on the basis of invariant sets under the time evolution in phase space. In particular, an attractor plays an essential role in dissipative systems because the asymptotic motion follows the attractor. However, when a value of control parameter is varied, an attractor or a part of it may change into a repeller, or a repeller may change into an attractor by bifurcations. A strange repeller is particularly important in understanding a chaotic transient, a presage of sustained chaos, the latter being caused by the strange attractor into which the former repeller changes. (An invariant set of saddle type is also called a repeller in this paper since it repels almost every point near it quickly or eventually.) Some changes are triggered by a collision of the attractor with a coexisting invariant set, which is a repeller in many cases. Moreover, there often exist two or more attractors for a fixed value of control parameter and their basins of attraction intermingle with each other in a very complicated manner. All these facts show that it is desirable that we take into consideration all the relevant invariant sets coexisting in an extended region of phase space. In the present paper, this will be done by extending the definition of the fluctuation spectrum of local expansion rates of nearby orbits to a global region of phase space. The exponential expansion of nearby orbits is an essential mechanism of chaos, and therefore this fluctuation spectrum is an important quantity characterizing chaotic states.

Crossover between Ballistic and Normal Diffusion

Crossover between ballistic motion and normal diffusion is studied based on the continuous-time random walk (CTRW) approach in order to analyze universal properties of strongly correlated motion and the decay process of correlation in deterministic diffusion. There exists a characteristic time scale r. For the time region t « T, ballistic motion is observed, which is followed by normal diffusion for t >> T. Higher-order moments are analytically obtained, and it is found that they obey scaling relations that are reminiscent of the generalized extended self-similarity (GESS) found in turbulent systems. As a simple dynamical system for numerical simulations, the climbing sine map in the vicinity of band crisis is considered. Good agreement between the theory and the numerical simulations is observed.

Continuous-Time Random Walk Approach to On-Off Diffusion

Statistical properties and scale invariances of on-off diffusion, which is an anomalous transport phenomenon caused by on-off intermittency, are studied on the basis of the continuous-time random walk (CTRW) approach. The anomalous production of heat is also analyzed. Scaling functions of the time evolution of the mean square displacement and the probability density function (PDF) of the position are analytically derived. It is found that there is a characteristic time separating two regimes of time intervals with different scaling laws for the PDF. In the interval that exists at times much smaller than the characteristic time, anomalous subdiffusion appears, which is followed by normal diffusion. In the earlier time interval, aside from the neighborhood of the origin, the PDF takes the form of a power law multiplied by a stretched exponential function, whereas in the later time interval, the PDF becomes a Gaussian. The results are compared with these model simulations. Good agreement between the theory and the simulation is obtained.

Anomalous Time Scaling of the Mean Square Distance in On-Off Diffusion

The mean square distance σ 2 ( t ) of the diffusion induced by on-off intermittency is derived based on the continuous-time random walk theory. It obeys a scaling law σ 2 ( t )=2 D t φ( t /τ) with diffusion constant D and characteristic time τ, which is confirmed by the use of numerical iterations of a specific periodic map. The scaling function φ and the power spectrum of the on-off intermittency variable I (ω) are analytically obtained from the distribution function of the laminar duration. Normal diffusion (φ( z )∼1) and slow diffusion (\(\phi(z)\propto 1/\sqrt{z}\)) are observed, respectively, for t ≫τ and t ≪τ. The former and latter correspond to the flat part ( I (ω)∼ c o n s t ) and the power law (\(I(\omega)\propto 1/\sqrt{\omega}\)), respectively, for the power spectrum.

Network Analysis Based on Statistical-Thermodynamics Formalism

Random walk on a graph is analyzed on the basis of the statistical-thermodynamics formalism to find phase transitions in network structure. Each phase can be related to a characteristic local structure of the network. For this purpose, the generalized transition matrix or the generalized Frobenius-Perron operator is introduced, whose largest eigenvalue yields statistical structure functions. The weighted visiting frequency related to the Gibbs probability measure, which turn out to be useful to extract characteristic local structures, is obtained from the inner product of the right and left eigenvectors corresponding to the largest eigenvalue. An algorithm to extract the characteristic local structure of each phase is also suggested based on this weighted visiting frequency.

Time Correlation Calculation Method Based on Delayed Coordinates

An approximate calculation method of time correlations by use of delayed coordinate is proposed. For a solvable piecewise linear hyperbolic chaotic map, this approximation is compared with the exact calculation, and an exponential convergence for the maximum time delay M is found. By use of this exponential convergence, the exact result for M →∞ is extrapolated from this approximation for the first few values of M . This extrapolation is shown to be much better than direct numerical simulations based on the definition of the time correlation function. As an application, the irregular dependence of diffusion coefficients similar to Takagi or Weierstrass functions is obtained from this approximation, which is indistinguishable from the exact result only at M = 2. The method is also applied to the

NETWORK AS A CHAOTIC DYNAMICAL SYSTEM

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius–Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure.