Tatsuo Yanagita

Coupled map lattice model for convection

Abstract A coupled map lattice model for convection is proposed, which consists of Eulerian and Lagrangian procedures. Simulations of the model reproduce wide-ranged phenomena in Benard convection experiments: For a small aspect ratio, formation of convective rolls, their oscillation, and many routes to chaos are found, with the increase of Rayleigh number. For a large aspect ratio, spatiotemporal intermittency is observed. For a high Rayleigh number, transition from soft to hard turbulence is fonfirmed, as is characterized by the temperature distribution change from Gaussian to exponential. The roll formation in a three-dimensional convection is also simulated, which reproduces experiments well.

Phenomenology of boiling: A coupled map lattice model

A minimal model for boiling is proposed. With increasing temperature of a bottom plate, the model shows three successive phases; conduction, nucleate, and film boiling. In the nucleate regime the heat flux increases with the temperature of the bottom plate, while it decreases in the film boiling regime. In the boiling phase, the maximum Lyapunov exponent is positive, implying that the boiling phenomena are spatiotemporally chaotic.

Chaotic Pulses for Discrete Reaction Diffusion Systems

Existence and dynamics of chaotic pulses on a one-dimensional lattice are discussed. Traveling pulses arise typically in reaction diffusion systems like the FitzHugh-Nagumo equations. Such pulses annihilate when they collide with each other. A new type of traveling pulse has been found recently in many systems where pulses bounce off like elastic balls. We consider the behavior of such a localized pattern on one-dimensional lattice, i.e., an infinite system of ODEs with nearest interaction of diffusive type. Besides the usual standing and traveling pulses, a new type of localized pattern, which moves chaotically on a lattice, is found numerically. Employing the strength of diffusive interaction as a bifurcation parameter, it is found that the route from standing pulse to chaotic pulse is of intermittent type. If two chaotic pulses collide with appropriate timing, they form a periodic oscillating pulse called a molecular pulse. Interaction among many chaotic pulses is also studied numerically.

Design of Oscillator Networks with Enhanced Synchronization Tolerance against Noise

Can synchronization properties of a network of identical oscillators in the presence of noise be improved through appropriate rewiring of its connections? What are the optimal network architectures for a given total number of connections? We address these questions by running the optimization process, using the stochastic Markov Chain Monte Carlo method with replica exchange, to design networks of phase oscillators with increased tolerance against noise. As we find, the synchronization of a network, characterized by the Kuramoto order parameter, can be increased up to 40%, as compared to that of the randomly generated networks, when the optimization is applied. Large ensembles of optimized networks are obtained, and their statistical properties are investigated.

Coupled map lattice model for boiling

Abstract A simple model for boiling is proposed. With increasing the temperature of a bottom plate, our minimal model shows three successive phases; conduction, nucleate and film boiling. In the nucleate regime the heat flux increases with the temperature of the bottom plate, while it starts to decrease in the film boiling regime. In the boiling state, the maximum Lyapunov exponent is positive, implying that the boiling phenomena are spatiotemporal chaos.

Phase-Reduction Approach to Synchronization of Spatiotemporal Rhythms in Reaction-Diffusion Systems

Reaction-diffusion systems can describe a wide class of rhythmic spatiotemporal patterns observed in chemical and biological systems, such as circulating pulses on a ring, oscillating spots, target waves, and rotating spirals. These rhythmic dynamics can be considered limit cycles of reaction-diffusion systems. However, the conventional phase-reduction theory, which provides a simple unified framework for analyzing synchronization properties of limit-cycle oscillators subjected to weak forcing, has mostly been restricted to low-dimensional dynamical systems. Here, we develop a phase-reduction theory for stable limit-cycle solutions of infinite-dimensional reaction-diffusion systems. By generalizing the notion of isochrons to functional space, the phase sensitivity function - a fundamental quantity for phase reduction - is derived. For illustration, several rhythmic dynamics of the FitzHugh-Nagumo model of excitable media are considered. Nontrivial phase response properties and synchronization dynamics are revealed, reflecting their complex spatiotemporal organization. Our theory will provide a general basis for the analysis and control of spatiotemporal rhythms in various reaction-diffusion systems.

Is a Two-dimensional Butterfly Able to Fly by Symmetric Flapping?

We consider whether two rigid plates (on a hinge) swinging completely symmetrically with respect to the up and down directions can produce enough lift for flight. By simulating a simple flapping motion in two-dimensional space using a discrete vortex method, we found a new type of symmetry-breaking mechanism that allows for the generation of sufficient lift to realize steady-flapping flight. The most important factor in determining the behavior of the model is the nature of the flow following the second downstroke, in which the wing produces significant lift through its interaction with the separation vortices.

Exploration of order in chaos using the replica exchange Monte Carlo method

A method for exploring unstable structures generated by non-linear dynamical systems is introduced. It is based on the sampling of initial conditions and parameters using the replica exchange Monte Carlo method, and it is efficient in searching for rare initial conditions and in the combined search for rare initial conditions and parameters. Examples discussed here include the sampling of unstable periodic orbits in chaos and searching for the stable manifold of unstable fixed points, as well as construction of the global bifurcation diagram of a map.

Monopoly, oligopoly and the Invisible Hand

Abstract We investigate the time evolution of the market structure, employing a discrete-time, nonlinear model characterized by agents with bounded rationality and product differentiation. By bounded rationality we mean that agents only have partial information: each firm does not know the demand function, so that it revises production decisions and prices so as to raise its profit based on the reaction by consumers. In this sense producers behave adaptively. We further assume product differentiation: each consumer has a preference for the product of a particular firm and exhibits habitual purchasing behavior unless price differences exceed a certain critical level. Simulation results show that monopoly and oligopoly emerges out of competitive situations as the key parameter of consumer’s inertia increases.

Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case in which the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.