Hirokazu Fujisaka

Stability Theory of Synchronized Motion in Coupled-Oscillator Systems

The general stability theory of the synchronized motions of the coupled· oscillator systems is developed with the use of the extended Lyapunov matrix approach. We give the explicit formula for a stability parameter of the synchronized state W unlfWhen the coupling strength is weakened, the coupled system may exhibit several types of non· synchronized motion. In particular, if W Unlf is chaotic, we always get a transition from chaotic Wunlf to a certain non· uniform state and finally the non·uniform chaos. Details associated with such transition are investigated for the coupled Lorenz model. As an application of the theory, we propose a new experimental method to directly measure the positive Lyapunov exponent of intrinsic chaos in reaction systems.

Statistical Dynamics Generated by Fluctuations of Local Lyapunov Exponents

Fluctuation-effect of local Lyapunov exponents on the distance between two nearby trajectories is studied from a statistical-mechanical point of view. It is shown that fluctuations of the logarithmic distance between two nearby trajectories are asymptotically described as the diffusion process, if the system is chaotic, obeying the normal distribution. The diffusion coefficient is proved to be invariant under conjugation. Onset behaviors of the diffusion near two typical transition points (the accumulation point of the Feigenbaum bifurcations and the Pomeau-Manneville intermittency transition) are briefly discussed.

Stability Theory of Synchronized Motion in Coupled-Oscillator Systems. III Mapping Model for Continuous System

By starting with a reaction-diffusion equation a mapping model for the continuous system is proposed. The transition from the uniform state to the non-uniform one occurs at the same value of the diffusion constant for the mapping model as for the original reaction-diffusion equation if the transition exists. The mapping model is further studied by adopting the logistic model in one-dimensional space with a periodic boundary condition. Equal time spectra in wave number space and power spectra for several values of wave numbers are numerically obtained. A comparison of the numerical results of the equal time spectra with a simple theory is made to give a satisfactory agreement for large wave numbers.

Theory of Diffusion and Intermittency in Chaotic Systems

On etudie la dynamique statistique de A t obeissant a A t+1 =B(x t )A t , ou B est une fonction de x t genere par une application chaotique a 1 dimension x t+1 =f(x t ). On montre que A t a 2 aspects complementaires; diffusion et intermittence

Breakdown of chaos symmetry and intermittency in the double-well potential system

Abstract The chaos-chaos transition in a one-particle system in the symmetric double-well potential under an external periodic field is studied from the viewpoint of the breakdown of the chaos symmetry and the development of the intermittency characteristics. It is found that the similarity exponent introduced to analyze the intermittency characteristics satisfies a scaling law near the transition point.

Intermittency Caused by Chaotic Modulation. II—Lyapunov Exponent, Fractal Structure and Power Spectrum—

On etudie un systeme application sous une modulation chaotique, qui a le point fixe, independamment du parametre de controle et de modulation

Theory of Diffusion and Intermittency in Chaotic Systems. III New Approach to Temporal Correlations

On presente une approche de mecanique statistique de la correlation temporelle dans les series temporelles a une dimension engendrees par une dynamique chaotique

Static and dynamic scaling laws near the symmetry-breaking chaos transition in the double-well potential system

Abstract The fluctuation of dynamics near the symmetry-breaking chaos transition in the double-well potential system under an external periodic excitation is studied with the fluctuation spectrum theory and the order- q time correlation function. The anomalous elongation of temporal correlation near the transition point, associated with the development of the intermittency characteristic, is shown to be described especially by the dynamic scaling law of the oder- q time correlation function.

Continued Fraction Expansion of Fluctuation Spectrum and Generalized Time Correlation

A practical approximation method for the fluctuation spectrum and generalized time correlation for a time series observed in stochastic or chaotic dynamics is proposed by utilizing the continued fraction expansion. The present approach enables us to evaluate the fluctuation spectrum and generalized time correlation in a systematic way without trying to find -exact solutions of the eigenvalue problem.

Breakdown of Chaos Symmetry and Intermittency in Band-Splitting Phenomena

A band·splitting phenomenon is studied from the viewpoint of the breakdown of the chaos symmetry and the development of the intermittency. The analyses are carried out for two kinds of coarse·grained variables relevant to the symmetry argument and the burst dynamics, mainly by utilizing similarity exponents introduced for analyzing one·dimensional time series. The develop· ment of the intermittency near the band·splitting point is shown to be chan.cterized as the scaling laws for the similarity exponents.