Dynamics in the anisotropic XY model driven by dichotomous Markov noise.

The statistics of a subcritical spatially homogeneous XY spin system driven by dichotomous Markov noise as an external field is investigated, particularly focusing on the switching process of the sign of the order parameter parallel to the external field. The switching process is classified in two types, which are called the Bloch-type switching and the Ising-type switching, according to whether or not the order parameter perpendicular to the external field takes finite value at the switching. The phase diagram for the onset of the switching process with respect to the amplitude of the external field and the anisotropy parameter of the system is constructed. It is revealed that the power spectral density I(omega) for the time series of the order parameter in the case of the Bloch-type switching is proportional to omega(-32) in an intermediate region of omega. Furthermore, the scaling function of I(omega) near the onset point of the Bloch-type switching is derived.

Domain-size statistics in the time-dependent Ginzburg-Landau equation driven by a dichotomous Markov noise.

The domain dynamics of magnetization obeying the time-dependent Ginzburg-Landau equation driven by a dichotomous Markov noise is discussed. The system with various domain sizes in the early stage temporally evolves following an annihilation of neighboring domain walls, where each domain wall moves diffusively. Three statistics on the domain size, i.e., average domain size, the ensemble average of the domain size distribution function, and the spatial power spectrum of the magnetization, are evaluated to characterize the domain wall annihilation process. A phenomenological evolution equation for the domain-size distribution function is constructed by simplifying the annihilation process of the domain wall appropriately, and the underlying mechanism of those statistics is investigated.

Periodic-orbit determination of dynamical correlations in stochastic processes.

It is shown that the large-deviation statistical quantities of the discrete-time, finite-state Markov process P_{n+1};{(j)}= summation _{k=1};{N}H_{jk}P_{n};{(k)} , where P_{n};{(j)} is the probability for the j state at the time step n and H_{jk} is the transition probability, completely coincide with those from the Kalman map corresponding to the above Markov process. Furthermore, it is demonstrated that, by using simple examples, time correlation functions in finite-state Markov processes can be well described in terms of unstable periodic orbits embedded in the equivalent Kalman maps.

Renormalized phase dynamics in oscillatory media.

Based on the complex Ginzburg-Landau equation (CGLE), a new mapping model of oscillatory media is proposed. The present dynamics is fully determined by an effective phase field renormalized by amplitude. The model exhibits phase turbulence, amplitude turbulence, and a frozen state reported in the CGLE. In addition, we find a state in which the phase and amplitude have spiral structures with opposite rotational directions. This state is found to be observed also in the CGLE. Thus, one concludes that the behaviors observed in the CGLE can be described by only the phase dynamics appropriately constructed.

Level dynamics approach to the large deviation statistical characteristic function.

We propose a level dynamics approach to the large deviation statistical characteristic function phi(q) for temporal series of dynamical variable V, which is the largest eigenvalue of the generalized evolution operator H{q}(identical with H+qV). This is done first by deriving "equations of motion" for the eigenvalues and the eigenstates of H{q} with the initial conditions determined by those of H , the true evolution operator for the dynamical variable under consideration, and then by solving these equations. Furthermore, utilizing simple solvable models, it is shown that the eigenvalues and eigenstates satisfy the equations of motion derived in this paper.

Dynamic phase transition in a rotating external field.

Dynamic phase transition in the Ginzburg-Landau model of the anisotropic XY spin system in a rotating external field is studied. We observe several types of oscillations, limit cycles, quasiperiodic oscillations and chaotic motions. It is found that limit cycle oscillations can have the periodicity of multiple times of the period of the applied field and that the system shows two kinds of scenarios leading to the onset of quasiperiodic oscillations, i.e., the saddle-node and Hopf bifurcations. Furthermore, this paper reports the findings of chaotic behaviors in the context of dynamic phase transition and that there exist two types of chaos with and without a certain kind of symmetry.

Irregular parameter dependence of generalized diffusion coefficients based on large deviation statistical analysis.

The nonperturbative non-Gaussian characteristics of diffusive motion are examined in the framework of the large deviation statistical theory, where simple extended mapping models showing chaotic diffusion are taken as an example. Furthermore, by rigorously solving the large deviation statistical quantities, it is found that the same type of anomalous, complex control parameter dependence as that for the diffusion coefficient reported by Klages and Dorfman is also observed in the large deviation statistical quantities such as the weighted average, the generalized diffusion coefficient, and the generalized power spectrum densities.

Critical dynamics of phase transition driven by dichotomous Markov noise.

An Ising spin system under the critical temperature driven by a dichotomous Markov noise (magnetic field) with a finite correlation time is studied both numerically and theoretically. The order parameter exhibits a transition between two kinds of qualitatively different dynamics, symmetry-restoring and symmetry-breaking motions, as the noise intensity is changed. There exist regions called channels where the order parameter stays for a long time slightly above its critical noise intensity. Developing a phenomenological analysis of the dynamics, we investigate the distribution of the passage time through the channels and the power spectrum of the order parameter evolution. The results based on the phenomenological analysis turn out to be in quite good agreement with those of the numerical simulation.

Domain dynamics in the anisotropic Swift-Hohenberg equation.

Two types of asymptotic ordering processes in the anisotropic Swift-Hohenberg equation are studied, paying particular attention to the interaction between domain walls. For the first type, we will discuss the time evolution in which the spatially oscillatory patterns are formed, and show that two kinds of patterns exist depending on whether or not the imaginary part of the field vanishes. When the imaginary part is present, the equation has two distinct states which are regarded as kinds of domains, so the dynamics between two domain walls is established. We then discuss, for the second type, the dynamics when nontrivial uniform states are constructed. There exist two different domain walls, the Ne el type wall and the Bloch type wall, in a similar way to the anisotropic Ginzburg-Landau equation. The equation of motion for two domain walls is derived, and it is shown that the distance between the two domain walls eventually approaches a finite length. The theoretical result is confirmed by numerical simulations. This fact proves the validity of the prediction on the temporal development of the distance between two domain walls.

Chaotic itinerancy in the oscillator neural network without Lyapunov functions.

Chaotic itinerancy (CI), which is defined as an incessant spontaneous switching phenomenon among attractor ruins in deterministic dynamical systems without Lyapunov functions, is numerically studied in the case of an oscillator neural network model. The model is the pseudoinverse-matrix version of the previous model [S. Uchiyama and H. Fujisaka, Phys. Rev. E 65, 061912 (2002)] that was studied theoretically with the aid of statistical neurodynamics. It is found that CI in neural nets can be understood as the intermittent dynamics of weakly destabilized chaotic retrieval solutions.

Magnetic walls in the anisotropic XY-spin system in an oscillating magnetic field.

Wall structures associated with dynamic phase transitions in the anisotropic XY -spin system in a temporally oscillating magnetic field h cos (Omegat) in a one-dimensional system are analyzed by using the time-dependent Ginzburg-Landau model. It is numerically confirmed that there exist two types of magnetic walls, i.e., the Néel and Bloch walls, and is found that the transition between the two walls can occur for changing h or Omega . The phase diagram for the stable regions of each wall is obtained by both numerical and analytical methods. Furthermore, the critical behavior of the modulus of the Bloch wall around the Néel-Bloch transition point is studied, and it is found that the transition can be either continuous or discontinuous with respect to h, depending on Omega .

Pattern dynamics associated with on-off convection in a one-dimensional system.

A numerical and theoretical analysis of the phenomenologically constructed nonlinear stochastic model of on-off intermittency experimentally observed by John et al. in the electrohydrodynamic convection in nematic liquid crystal under applied dichotomous electric field is carried out. The model has the structure of the one-dimensional Swift-Hohenberg equation with a fluctuating threshold which represents an applied electric field and either with or without additive noise which corresponds to thermal noise. It is found that the fundamental statistics of pattern dynamics without additive noise agree with those experimentally observed, and also with those reported previously in two-dimensional system. In contrast to that the presence of multiplicative noise generates an intermittent evolution of pattern intensity, whose statistics are in agreement with those of on-off intermittency so far known, the additive noise gives rise to the change of position of the convective pattern. It is found that the temporal evolution of the phase suitably introduced to describe the global convective pattern also shows an intermittent evolution. Its statistics are studied in a detailed way with numerical simulation and stochastic analysis. The comparison of these results turn out to be in good agreement with each other.

Synchronization and intermittency in three-coupled chaotic oscillators.

Synchronization of three-coupled chaotic oscillators was studied with the use of a coupled map system derived for interacting kicked relaxators. Partial synchronization (PS), in which two of the three were synchronized, was observed in addition to complete synchronization. An intermittency associated with the breakdown of the PS, seemingly different from the conventional on-off intermittency, was found. We elucidated the statistics, observing the burst-size distribution, the laminar duration distribution, etc. It was found that the breakdown of the PS generated an anomalous diffusion different from that associated with on-off intermittency.

Intermittency and exponent field dynamics in developed turbulence.

Spatiotemporal dynamics of intermittency in association with coarse-grained energy-dissipation rate fluctuations is discussed. This is done first by phenomenologically constructing the probability density for exponent field fluctuations that is introduced to characterize the energy-dissipation rate field, and then by proposing the Langevin dynamics derived with the projection-operator method on the basis of the Navier-Stokes equation. With a Gaussian approximation for exponent fluctuations, spatiotemporal correlation functions for coarse-grained energy-dissipation rate fluctuations are explicitly obtained.

Dynamic phase transitions in the anisotropic XY spin system in an oscillating magnetic field.

The Ginzburg-Landau model for the anisotropic XY spin system in an oscillating magnetic field below the critical temperature T(c), psi;(r,t)=(T(c)-T)psi-/psi/(2)psi+gammapsi(*)+ nabla (2)psi+h cos(Omegat) is both theoretically and numerically studied. Here psi is the complex order parameter and gamma stands for the real anisotropy parameter. It is numerically shown that the spatially uniform system shows various characteristic oscillations (dynamical phases), depending on the amplitude h and the frequency Omega of the external field. As the control parameter, either h or Omega, is changed, there exist dynamical phase transitions (DPT), separating them. By making use of the mode expansion analysis, we obtain the phase diagrams, which turn out to be in a qualitative agreement with the numerically obtained ones. By carrying out the Landau expansion, the reduced equations of motion near the DPT are derived. Furthermore, taking into account the spatial variation of order parameters, we will derive the analytic expressions for domain walls, which are represented by the Néel and Bloch type walls, depending on the difference of the coexistence of phases.

Stability of oscillatory retrieval solutions in the oscillator neural network without Lyapunov functions.

Constructing a Ginzburg-Landau map neural network, we analyze its storage capacity with an equilibrium theory of the self-consistent signal-to-noise analysis (SCSNA); however, the prediction does not consist with the simulation results just in the parameter region where the characteristic of the non-Lyapunov-function system gets enhanced [J. Phys. A 32, 4623 (1999)]. It is expected that this inconsistency comes from the fact that the dynamics of retrieval and nonretrieval states governs the phase transition. Alternatively, we investigate its storage capacity with the help of the Amari-Maginu-Okada theory, a dynamical theory, for the stability analysis of dynamical states. We consequently found that the theory predicts dynamical states quite well especially in the region where the SCSNA breaks down, and that the phase diagram coincides quantitatively well with the simulation results.

Scaling hypothesis leading to generalized extended self-similarity in turbulence.

A scaling hypothesis leading to generalized extended self-similarity (GESS) for velocity structure functions, valid for intermediate scales in isotropic, homogeneous turbulence, is proposed. By introducing an effective scale r, monotonically depending on the physical scale r, with the use of the large deviation theory, the asymptotic forms of the probability densities for the velocity differences u(r) and for the coarse-grained energy-dissipation rate fluctuations epsilon(r), compatible with this GESS, are proposed. The probability density for epsilon(r) is shown to have the form P(r)(epsilon) approximately equal to epsilon(-1)(r/L)(S(r)[z(r)](epsilon))) with z(r)(epsilon)=ln(epsilon/epsilon(L))/ln(L/r), where L and epsilon(L) are the stirring scale and the coarse-grained energy-dissipation rate over the scale L. The concave function S(r)(z), the spectrum, plays the central role of the present approach. Comparing the results with numerical and experimental data, we explicitly obtain the fluctuation spectra S(r)(z).

Universality of chaotic rare fluctuations in a locally coupled phase map model.

Chaotic fluctuations of the order parameter in a coupled two-dimensional phase map model are numerically investigated. We discuss the system-size N dependence of the statistical properties of rare fluctuations observed in the transition range between the quasiordered chaotic state and the fully developed one. It is found that the normalized probability distribution function (PDF) has a unique functional form irrespective of N. The asymptotic form of the PDF is discussed in connection with the universal distribution for correlated systems proposed by Bramwell et al. [Nature (London) 396, 552 (1998)]. Moreover, it is observed that the power spectrum P(N)(omega) of rare fluctuations asymptotically takes the power-law form P(N)(omega) equivalent to omega(-(1+alpha)) (alpha=0.6 equivalent to 0.7) irrespective of N. This result suggests that the temporal correlation decays as a stretched exponential.