Shinji Koga

A variety of stable persistent waves in intrinsically bistable reaction-diffusion systems: from one-dimensional periodic waves to one-armed and two-armed rotating spiral waves

Abstract Various types of non-trivial stable persistent waves in intrinsically bistable reaction-diffusion equations are studied with an emphasis on initiation problems of these waves. The reaction-diffusion equation of Rinzel and Kellers type allows us to obtain exact plane wave solutions and their dispersion relation curves with three branches in most cases. Coexistence of six plane wave solutions is also discussed. On the basis of the knowledge of the stable plane wave solutions, the initiation problem for multi-armed rotating spiral waves is discussed. Numerical simulations show that a one-armed spiral wave maintains its wave pattern, and that a two-armed spiral wave survives for a long time before it eventually dies out.

Coexistence of stably propagating periodic wave trains in intrinsically bistable reaction-diffusion systems

Abstract The coexistence of stably propagating periodic waves is found in an intrinsically bistable reaction-diffusion equation without any external forces. The McKean model with bistable reaction terms allows us to discuss coexisting exact solutions which reside on neither of two fixed points.

Phase Description Method to Time Averages in the Lorenz System

On the basis of the transformation to the rotating coordinates associated with the imbedded unstable limit cycles in the Lorenz system, we present a new representation for the long time averages, which may be applicable to any three·dimensional dissipative dynamical systems producing chaos. By employing the dynamics of the phases of the imbedded limit cycles, we show that the time average is expressible in terms of two types of the weight factors; the residence time probability density and the factor inversely proportional to the speed of the phase.

GLOBAL SCALING LAWS IN 3D SINAI'S BILLIARDS

We find numerical evidence for the scaling laws of the moments of the free path length in 3D Sinais billiard systems with a simple hexahedron lattice involving an ellipsoid scatterer. We also find scaling exponents of the moments by regarding the mean free path length as the relevant scaling variable.

Scaling Laws of Moments in Sinai's Billiard Systems. I The Case of Rectangular Lattice

We investigate Sinais billiard system in 2 dimensions with an emphasis on the scaling laws of the moments of the free path length in the case of the rectangular lattice. By adopting the mean free path as the scaling variable, the moments of the non· integer power expressed by the integral formula are governed by the scaling laws. The scaling exponents tum out to be almost independent of the manner in which we change the ellipse·shaped scatterer. We also discuss the singularity problem of the free path length associated with the problem of whether the moments exist or not.

An approximation formula and parameter-dependence of statistical quantities in low-dimensional chaotic systems

We derive an approximation formula for obtaining parameter dependence of statistical quantities in chaotic systems on the basis of a Frobenius-Perron equation. The formula is characterized by three integers; One is the number of application of the Frobenius-Perron operator, the second is the number of the delta-peaks of the initial density, and the third is the number of the integration domains. We find numerically that this formula is applicable to a wide variety of states of a system ranging from stable cycles to band-chaos and totaly spread chaotic regime by merely changing the system parameter, except for the very narrow windows representing stable cycles with large periodicity, where the concrete examples are a logistic map, a Henon map and so on. We finally consider how we extend our theory to ODE’s.