Munehisa Sekikawa

Three-dimensional tori and Arnold tongues

This study analyzes an Arnold resonance web, which includes complicated quasi-periodic bifurcations, by conducting a Lyapunov analysis for a coupled delayed logistic map. The map can exhibit a two-dimensional invariant torus (IT), which corresponds to a three-dimensional torus in vector fields. Numerous one-dimensional invariant closed curves (ICCs), which correspond to two-dimensional tori in vector fields, exist in a very complicated but reasonable manner inside an IT-generating region. Periodic solutions emerge at the intersections of two different thin ICC-generating regions, which we call ICC-Arnold tongues, because all three independent-frequency components of the IT become rational at the intersections. Additionally, we observe a significant bifurcation structure where conventional Arnold tongues transit to ICC-Arnold tongues through a Neimark-Sacker bifurcation in the neighborhood of a quasi-periodic Hopf bifurcation (or a quasi-periodic Neimark-Sacker bifurcation) boundary.

ANALYSIS OF TORUS BREAKDOWN INTO CHAOS IN A CONSTRAINT DUFFING VAN DER POL OSCILLATOR

The bifurcation structure of a constraint Duffing van der Pol oscillator with a diode is analyzed and an objective bifurcation diagram is illustrated in detail in this work. An idealized case, where the diode is assumed to operate as a switch, is considered. In this case, the Poincare map is constructed as a one-dimensional map: a circle map. The parameter boundary between a torus-generating region where the circle map is a diffeomorphism and a chaos-generating region where the circle map has extrema is derived explicitly, without solving the implicit equations, by adopting some novel ideas. On the bifurcation diagram, intermittency and a saddle-node bifurcation from the periodic state to the quasi-periodic state can be exactly distinguished. Laboratory experiment is also carried out and theoretical results are verified.

Qualitative-Modeling-Based Silicon Neurons and Their Networks.

The ionic conductance models of neuronal cells can finely reproduce a wide variety of complex neuronal activities. However, the complexity of these models has prompted the development of qualitative neuron models. They are described by differential equations with a reduced number of variables and their low-dimensional polynomials, which retain the core mathematical structures. Such simple models form the foundation of a bottom-up approach in computational and theoretical neuroscience. We proposed a qualitative-modeling-based approach for designing silicon neuron circuits, in which the mathematical structures in the polynomial-based qualitative models are reproduced by differential equations with silicon-native expressions. This approach can realize low-power-consuming circuits that can be configured to realize various classes of neuronal cells. In this article, our qualitative-modeling-based silicon neuron circuits for analog and digital implementations are quickly reviewed. One of our CMOS analog silicon neuron circuits can realize a variety of neuronal activities with a power consumption less than 72 nW. The square-wave bursting mode of this circuit is explained. Another circuit can realize Class I and II neuronal activities with about 3 nW. Our digital silicon neuron circuit can also realize these classes. An auto-associative memory realized on an all-to-all connected network of these silicon neurons is also reviewed, in which the neuron class plays important roles in its performance.

REVEALING THE TRICK OF TAMING CHAOS BY WEAK HARMONIC PERTURBATIONS

Taming chaos by weak harmonic perturbations has been a hot topic in recent years. In this paper, the authors investigate a scenario for the mechanism of taming chaos via bifurcation theory, and ass...

Experimental study of complex mixed-mode oscillations generated in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation.

Bifurcations of complex mixed-mode oscillations denoted as mixed-mode oscillation-incrementing bifurcations (MMOIBs) have frequently been observed in chemical experiments. In a previous study [K. Shimizu et al., Physica D 241, 1518 (2012)], we discovered an extremely simple dynamical circuit that exhibits MMOIBs. Our model was represented by a slow/fast Bonhoeffer-van der Pol circuit under weak periodic perturbation near a subcritical Andronov-Hopf bifurcation point. In this study, we experimentally and numerically verify that our dynamical circuit captures the essence of the underlying mechanism causing MMOIBs, and we observe MMOIBs and chaos with distinctive waveforms in real circuit experiments.

An integrated circuit design of a silicon neuron and its measurement results

A novel MOSFET-based silicon nerve membrane model and its measurement results are described in this paper. This model is designed based on a mathematical structure that is characterized by phase plane analysis and bifurcation theory. The circuit is fabricated through MOSIS TSMC 0.35 μm CMOS process. Measurement results demonstrate that our circuit shows fundamental abilities of excitable cells such as a) a resting state, b) an action potential, c) a threshold, and d) a refractoriness.

A study on power device loss of DC-DC buck converter with SiC schottky barrier diode

Silicon carbide (SiC) is an expected candidate as a material for the next generation power semiconductor devices. In this paper, the authors evaluate the power device loss of a DC-DC buck converter, where the silicon (Si) free wheeling diode is replaced with an SiC schottky barrier diode (SBD). The loss reduction of the DC-DC buck converter is discussed with respect to switching frequency and duty ratio of the gate drive signal.

Chaos computing: a unified view

Chaos computing is a non-traditional new paradigm that exploits the extreme non-linearity of chaotic systems. This article presents a unified theoretical view of chaos computing. It introduces the fundamental concept and the unique features that are characteristics of chaos computing, and discusses various implementation approaches. Basic aspects of digital chaos computing to realise logical gates are introduced, followed by two specific techniques: (1) direct utilisation of the threshold mechanisms; (2) an application of the chaos neuron model. After presenting these approaches, we discuss general characteristics of digital chaos computing. Other digital, analog and digital/analog hybrid forms of chaos computing are also considered. Potential advantages of chaos computing include: high speed, low power and low cost, a general-purpose form of computing, re-configurable or dynamic logical architecture, implementation of continuous logic, robustness against noise, and parallel and distributed computing.

Nonlinear analysis on purely mechanical stabilization of a wheeled inverted pendulum on a slope

This paper investigates the potential for stabilizing an inverted pendulum without electric devices, using gravitational potential energy. We propose a wheeled mechanism on a slope, specifically, a wheeled double pendulum, whose second pendulum transforms gravity force into braking force that acts on the wheel. In this paper, we derive steady-state equations of this system and conduct nonlinear analysis to obtain parameter conditions under which the standing position of the first pendulum becomes asymptotically stable. In this asymptotically stable condition, the proposed mechanism descends the slope in a stable standing position, while dissipating gravitational potential energy via the brake mechanism. By numerically continuing the stability limits in the parameter space, we find that the stable parameter region is simply connected. This implies that the proposed mechanism can be robust against errors in parameter setting.

Complex Bifurcation of Arnol’d Tongues Generated in Three-Coupled Delayed Logistic Maps

This study investigates quasi-periodic bifurcations and Arnol’d resonance webs generated in a three-coupled delayed logistic map. Complex bifurcation structure is generated when a conventional Arnol’d tongue transits to a higher-dimensional Arnol’d tongue. We discovered that, at least, two periodic attractors coexist in the conventional Arnol’d tongue which can bifurcate to two one-tori via doubly-folded Neimark–Sacker bifurcation.