Yoichiro Takahashi

Chaos, External Noise and Fredholm Theory

Y oshitsugu OONO*l and Y oichiro TAKAHASHI* Research Institute of Industrial Science Kyushu University, Fukuoka 812 *Department of Afathematics, College of General Education University of Tokyo, Komaba, Tokyo 153 (Received December 7, 1979) The Fredholm theory of integral equations is applied to the Pcrron-Frobenius equation which determines the invariant measure of nonlinear difference equations. The resultant Fredholm determinant D is identical to 1/(, where ( is the Artine-Mazur-Ruelle (-function. From the investigation of D we get the observable condition of the invariant measure of chaos, its stability against external noise, etc.

Entropy Functional (free energy) for Dynamical Systems and their Random Perturbations

Publisher Summary This chapter discusses the entropy functional for dynamical systems and their random perturbations. The chapter formulates the Gibbs variational principle that unifies the Donsker-Varadhan theory for Markov chains, the equilibrium classical statistical mechanics of lattice systems and the theories for symbolic dynamics, expanding maps and Anosov diffeomorphisms. The chapter discusses the relation between the quantities for dynamical systems and the non-random limits of the corresponding quantities given for their random perturbations. The case of symbolic dynamics is also discussed. It is shown in the typical cases that the two functionals f and q coincide with each another and the result of the computation of them may be interpreted as a characterization of the metrical entropy. There is a discussion about the case of general maps of intervals. The counterexample leads to the conjecture that the entropy functional f is affine when the underlying dynamical system ( X , F ) is structurally stable. In a special case this conjecture is proved to be affirmative and the structurally stable systems are Morse-Smale systems.

Information flow in heterogeneously interacting systems

Motivated by studies on the dynamics of heterogeneously interacting systems in neocortical neural networks, we studied heterogeneously-coupled chaotic systems. We used information-theoretic measures to investigate directions of information flow in heterogeneously coupled Rössler systems, which we selected as a typical chaotic system. In bi-directionally coupled systems, spontaneous and irregular switchings of the phase difference between two chaotic oscillators were observed. The direction of information transmission spontaneously switched in an intermittent manner, depending on the phase difference between the two systems. When two further oscillatory inputs are added to the coupled systems, this system dynamically selects one of the two inputs by synchronizing, selection depending on the internal phase differences between the two systems. These results indicate that the effective direction of information transmission dynamically changes, induced by a switching of phase differences between the two systems.

Information Theoretic Approach to Dynamical Systems of Heterogeneously Interacting Chaotic Oscillators

We studied heterogeneously‐coupled Rossler oscillators in order to understand dynamics of hetero‐interactions. By using an information‐theoretic measure called transfer entropy, the directions of information flows in hetero‐interacting systems were investigated. We found the asymmetry of information flow emerged in the mutually‐coupled systems. Furthermore, the dominant direction of the information flow switched spontaneously in an intermittent manner, depending on the phase lag.

Classification of chaos and a large deviation theory for compact dynamical systems

A survey will be given on the large deviation theory aspect of the classification of chaos with a slight generalization of previous results.