Toru Ohira

Limit Cycles, Tori, and Complex Dynamics in a Second-Order Differential Equation with Delayed Negative Feedback

We analyze a second-order, nonlinear delay-differential equation with negative feedback. The characteristic equation for the linear stability of the equilibrium is completely solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The bifurcations occurring as the linear stability is lost are investigated by the construction of a center manifold: The nature of Hopf bifurcations and more degenerate, higher-codimension bifurcations are explicitly determined.

Unstable dynamical systems: Delays, noise and control

Escape from an unstable fixed point in a time-delayed dynamical system in the presence of additive white noise depends on both the magnitude of the time delay, τ, and the initial function. In particular, the longer the delay the smaller the variance and hence the slower the rate of escape. Numerical simulations demonstrate that the distribution of first passage times is bimodal, the longest first passage times are associated with those initial functions that cause the greatest number of delayed zero crossings, i.e. instances where the deviations of the controlled variable from the fixed point at times t and t- τ have opposite signs. These observations support the utility of control strategies using pulsatile stimuli triggered only when variables exceed certain thresholds.

Predictability of currency market exchange

We analyze tick data of yen–dollar exchange with a focus on its up and down movement. We show that there exists a rather particular conditional probability structure with such high frequency data. This result provides us with evidence to question one of the basic assumptions of the traditional market theory, where such bias in high frequency price movements is regarded as not present. We also construct systematically a random walk model reflecting this probability structure.

A dynamical structure of high frequency currency exchange market

We analyze tick-by-tick data, the most high frequency data available, of yen–dollar currency exchange rates. We show that a dynamical structure can be observed in binarized data indicating the direction of up and down movement of prices, which is not apparently seen from the price change itself. This result is consistent with our previous study that there exists a conditional probabilistic structure in binarized data. The dynamical and probabilistic structure which we found could indicate that dealers’ decision making is based on a binary strategy, even if they are unconscious of this fact.

Delayed Random Walks: Investigating the Interplay Between Delay and Noise

A model for a 1-dimensional delayed random walk is developed by generalizing the Ehrenfest model of a discrete random walk evolving on a quadratic, or harmonic, potential to the case of non-zero delay. The Fokker-Planck equation derived from this delayed random walk (DRW) is identical to that obtained starting from the delayed Langevin equation, i.e. a first-order stochastic delay differential equation (SDDE). Thus this DRW and SDDE provide alternate, but complimentary ways for describing the interplay between noise and delay in the vicinity of a fixed point. The DRW representation lends itself to determinations of the joint probability function and, in particular, to the auto-correlation function for both the stationary and the transient states. Thus the effects of delay are manisfested through experimentally measurable quantities such as the variance, the correlation time, and the power spectrum. Our findings are illustrated through applications to the analysis of the fluctuations in the center of pressure that occur during quiet standing.

Delayed stochastic resonance with a 1-D ring

We study here a simple one-dimensional ring composed of stochastic binary elements with delayed interaction. With this model we can analytically compute “interspike” interval histograms, and show how “resonance” or rhythmic behaviors arise with only noise and delay.

Visuomotor Tracking Tasks With Delayed Pursuit and Escape

Virtual stick balancing (VSB) is a manual visuomotor tracking task that involves interplay between a human and a computer in which the movements are programmed to resemble those of balancing a stick at the fingertip. Since time delays and random perturbations (“noise”) are intrinsic properties of this task, we modeled VSB as a delayed pursuit-escape process: the target movements are described by a simple random walk and those movements controlled by the computer mouse by a delayed random walk biased towards the target. As subjects become more skilled, a stereotyped and recurring pursuit‐escape pattern develops in which the mouse pursues the target until it overtakes it, causing the target to move in a different direction, followed, after a lag, by the pursing mouse. The delayed pursuit-escape random walk model captured the qualitative nature of this tracking task and provided insights into why this tracking task always fails at some point in time, even for the most expert subjects.

Stochasticity and non-locality of time

We present simple classical dynamical models to illustrate the idea of introducing a stochasticity with non-locality into the time variable. For stochasticity in time, these models include noise in the time variable but not in the “space” variable, which is opposite to the normal description of stochastic dynamics. Similarly with respect to non-locality, we discuss the delayed and predictive dynamics which involve two points separated on the time axis. With certain combinations of fluctuations and non-locality in time, we observe a “resonance” effect. This is an effect similar to stochastic resonance, which has been discussed within the normal context of stochastic dynamics, but with different mechanisms. We discuss how these models may be developed to fit a broader context of generalized dynamical systems where fluctuations and non-locality are present in both space and time.

Stochastic three-state model with delay

Abstract We investigate here a stochastic three-state element whose transition probability depends on its state at a fixed interval in the past. A rich dynamical behavior which does not appear in a binary model is observed in this model. Axa0resonant phenomenon due to the delay appears as well. Furthermore, an analytical expression of this resonant characteristic can be derived in general for an N -state model. We also discuss its application to the dynamical time series of yen–dollar currency exchange.

A Neural Network Model with Bidirectional Whitening

We present here a new model and algorithm which performs an efficient Natural gradient descent for multilayer perceptrons. Natural gradient descent was originally proposed from a point of view of information geometry, and it performs the steepest descent updates on manifolds in a Riemannian space. In particular, we extend an approach taken by the “Whitened Neural Networks” model. We make the whitening process not only in the feed-forward direction as in the original model, but also in the back-propagation phase. Its efficacy is shown by an application of this “Bidirectional Whitened Neural Networks” model to a handwritten character recognition data (MNIST data).