Hiroyuki Nagashima

Experiment on the Toda Lattice Using Nonlinear Transmission Lines

A nonlinear transmission line equivalent to the exponential lattice of Toda was obtained with the proper understanding of nonlinear properties of a capacitor used in a shunt branch. The character of a soliton is observed to follow the prediction given by Toda with respect to the shape and velocity of the soliton. The details of interaction of two solitons moving in the opposite or same direction are also found to be in good agreement with the theory. The dissipation effect on the soliton due to the small losses in the passive elements of the network is discussed in the continuum limit.

Strange Attractor of Chaotic Magnons Observed in Ferromagnetic (CH3NH3)2CuCl4

By parallel pumping experiments in ferromagnetic (CH 3 NH 3 ) 2 CuCl 4 chaotic auto-oscillations of magnon amplitude are observed at the pumping frequency of 9.39 GHz and the temperature of 1.65 K. Strange attractors constructed in three-dimensional phase space from time series data of chaotic oscillations display properties of stretching and folding of the sheet of trajectories. Fractal dimensions of strange attractors are obtained by the method of the correlation integral as a function of driving microwave power. With increasing driving power fractal dimension increases from 1.6 to 3.4. The largest Lyapunov exponent is obtained as 0.43±0.05 by using empirical return maps. The chaotic state arises directly from period-two oscillations without cascade of period doubling bifurcations.

Computer Simulation of Solitary Waves of the Nonlinear Wave Equation ut+uux-γ2u5x=0

Computer simulation of the nonlinear wave equation u t + u u x -γ 2 u 5 x =0 was carried out. The results show that one solitary wave with oscillatory tails propagates stably and it is described as u =λ f { λ 1/4 ( x -λ t )}. It is found that the two-solitary wave interaction is classified into two types, T and B, according to the relative amplitudes of waves, and after type the T interaction, both identities of solitary waves before the interaction are conserved, while after type the B interaction their identities are approximately conserved. Formation of a two-solitary waves bound state is observed after three-solitary wave interaction, and the condition of the bound state formation is discussed.

Chaotic States in the Belousov-Zhabotinsky Reaction

Nonperiodic oscillations have been observed in the Belousov-Zhabotinsky reaction in a closed and well-stirred system. By plotting each amplitude (or peak-to-peak value) against the preceding one (Lorenz plot), we obtained well defined single-valued transition functions which allow for a period three. Thus, in the sense of Li and York, we conclude that the system exhibits chaotic behavior.

Chaos in a nonlinear wave equation with higher order dispersion

Abstract Chaos in the equation ut + uux + αu3x−βu5x = 0 has been studied. The largest Lyapunov number in an infinite-dimens ional amplitude space is computed. It is found that the chaos originates because the solitary waves do not conserve their identity in a collision.

Target Patterns and Pacemakers in a Reaction-Diffusion System

An experimental study on pacemakers of target patterns in the oscillatory Belousov-Zhabotinsky reaction is carried out. It is found that wavelength λ of the pattern is controllable by changing the radius r of the pacemaker and that the obtained relationship between λ and r can be qualitatively explained by a simple equation based on a perturbational approach to the system.

Experiment on Chaotic Responses of a Forced Belousov-Zhabotinsky Reaction

The periodic perturbation is imposed on the Belousov-Zhabotinsky reaction by means of periodically repeated stirring of the reacting mixture. Since the flow of the mixture caused by the stirring includes oxygen gas from the surface, the reaction is modulated by a periodic stirring and an entrained or a nonperiodic oscillation is observed depending on the repetition rate of the stirring. The sample data amplitudes of nonperiodic oscillations are analysed by using the Lorenz plot, and well-defined single-valued transfer functions are obtained which give positive Liapounov numbers and have period three points. We conclude thus that the system exhibits chaotic behavior.

Map-Based Analysis of Noise-Induced Convergence in Chaos

The mechanism and condition of appearance of order in chaos with Gaussian white noise is investigated based on the Poincare return map constructed from the three-variables ODE of the Belousov-Zhabotinsky reaction. When the noise is added to the chaos having the bifurcation parameter near period-three oscillation, it occasionally happens that topological entropy is constant but the Kolmogorov entropy decreases. Analysis on three-times iterated map reveals that this phenomenon, named “noise-induced convergence”, is caused by an increase of the length of laminar phase and the subsequent change of the invariant density. The same phenomenon is observed in m-times iterated map of the chaos of the logistic map. We consider that “noise-induced convergence” is characteristic of intermittent chaos.

Ring-Shaped Model of the Pacemaker in Oscillatory Reaction-Diffusion System.

This paper presents a new model of a pacemaker in the oscillatory reaction-diffusion system called “the ring-shaped model”. The present model has a frequency higher than that of the bulk oscillations in a ring-shaped region. Theoretical and experimental studies have revealed that the “ring-shaped model” explains the experiments with a piece of disk-shaped filter paper as a pacemaker, indicating that the wavelength is a function of the radius of the pacemaker and has a minimum value.

Effect of Noise on Chaos in a One-Dimensional Map

The mechanism and the characteristics of appearance of order in chaos with Gaussian white noise are investigated in the logistic map. This order (noise-induced order) appears as the constancy of the topological entropy and the decrease of the Kolmogorov entropy. Analysis of n-times iterated map reveals that “noise-induced order” is caused by the increase of the length of the laminar region and the subsequent change of the invariant density, where “local structure” overcomes “global structure”. We find that “noise-induced order” is accelerated by an increase of the noise period, and takes place only in chaos near period-five and -under windows.