The term “bifurcation” means changes in qualitative structure of dynamical systems due to a small change of the parameter values. In the engineering systems such as communication, circuits and control systems, the bifurcation relates deeply to border of stability and generation of complicated nonlinear phenomena including chaos. There exist various kinds of bifurcation phenomena. Classification and recognition of the phenomena are very important to understand the nonlinear dynamics and to develop their efficient applications. This special section consists of two invited papers and ten regular papers. They have valuable contents which are important to grasp the state-of-the-art of the study and to trigger further development of the study. The first invited paper (Tsumoto, Ueta, Yoshinaga and Kawakami) discusses fundamental theory and computation methods of bifurcation analysis in nonlinear dynamical systems. The second invited paper (In, Kho, Longhini and Palacios) discusses global bifurcation of coupled overdamped bistable systems with applications to sensor devices. The ten regular papers discuss a variety of topics: a visualization method for chaotic attractors in planer discrete systems, numerical sensitivity in the analysis of a high-dimensional oscillator, bifurcation of a piecewise linear 1D map of a simple hybrid dynamical system, analysis of super-stable periodic orbits of 1D map with a trapping window, bifurcation of an interrupted dynamical system with a periodic threshold, bifurcation scenarios of inhibitory responses in a simple spiking neuron model, dependence of parameters on chaos-based solvers of combinatorial optimization problems, the particle swarm optimization algorithm for designing the class-E amplifier, basic analysis of digital spike maps and a cellular array system for parallel generation of pseudo-random binary sequences. On behalf of the editorial board, this guest editor expresses his sincere thanks to all the authors for their excellent contributions. He also thanks the reviewers and the members of the editorial board, especially, Assoc. prof. Hiroyuki Torikai of Osaka University and Assoc. prof. Takuji Kosaka of Oita University for their supports on publishing this special section. It may be some destiny that this special section is published on the 100th anniversary of the death of Henri Poincaré who introduced the term “bifurcation” in 1885.
In this paper, we explain how to compute bifurcation parameter values of periodic solutions for non-autonomous nonlinear differential equations. Although various approaches and tools are available for solving this problem nowadays, we have devised a very simple method composed only of basic computational algorithms appearing in textbooks for beginner's, i.e., Newton's method and the Runge-Kutta method. We formulate the bifurcation problem as a boundary value problem and use Newton's method as a solver consistently. All derivatives required in each iteration are obtained by solving variational equations about the state and the parameter. Thanks to the quadratic convergence ability of Newton's method, accurate results can be quickly and effectively obtained without using any sophisticated mathematical library or software. If a discontinuous periodic force is applied to the system, we can use the same strategy to solve the bifurcation problem. The key point of this method is deriving a differentiable composite map from the various information about the problem such as the location of sections, the periodicity, the Poincaré mapping, etc.
Theoretical and experimental works have shown that coupling similar overdamped bistable systems can lead, under certain conditions that depend on the topology of connections and the number of units, to self-induced large-amplitude oscillations that emerge through a global bifurcation of heteroclinic connections between saddle-node equilibria. We have exploited this fundamental feature to model, design, and fabricate a new generation of highly-sensitive, low-powered, sensor devices, mainly detectors of magnetic- and electric-fields. In this article we review the fundamental principles and methods behind this new paradigm. These principles are device-independent so they can be readily adapted to a wide range of sensors, such as acoustic and gyroscopic sensors among many types. In this manuscript we describe the design and fabrication of a new class of highly-sensitive fluxgate- and electric-field magnetometers as a case study to review basic ideas and methods.
We propose a visualization method called the directional coloring for chaotic attractors in planer discrete systems. A color in the hue circle is assigned to the argument determined by the current point and its n-th mapped point. Some unstable n-periodic points embedded in the chaotic attractor become visible as radiation points and they can be accurately detected by combination of this coloring and the Newton's method. For a chaotic attractor in a non-invertible map, we find out invariant patterns around the fixed point and detect its nearest unstable n-periodic point. The computed results of their locations show a fractal property of the system.
We have encountered serious difficulties when we analyze bifurcation phenomena in a high-dimensional oscillator by Lyapunov analysis. Our model uses an eight-dimensional piecewise-linear oscillator containing a hysteresis element. The numerical results show that the solutions do not arrive at a stationary state quite often even if we use piecewise-linear exact solutions and remove more than 100,000 transient iteration of the Poincaré map. Furthermore, we calculate the Lyapunov exponents by averaging 1,000,000 iteration of the Jacobian matrix of the Poincaré map after removing transient 500,000 iteration. The values of the Lyapunov exponents themselves could be trustworthy because piecewise-linear exact solutions have been employed. However, it is very difficult to classify the solutions from the value of the calculated Lyapunov exponents even if the attractor is not chaotic. This is because the bifurcation structure of a higher-dimensional torus is extremely complicated. In some cases, it is very hard to settle how many iteration is necessary to calculate reasonable Lyapunov exponents, and to distinguish whether Lyapunov exponents are exactly zero or not from the values of the numerically calculated Lyapunov exponents. This is a major concern of this study. These complex situations are spotlighted by observing attractors and drawing the graph of the Lyapunov exponents.
A very simple hybrid circuit proposed as chaos generator is studied. It is modeled using a 3-slopes piecewise linear map defined on [0,1] and depending upon three parameters. The parameter space is investigated in order to classify regions of existence of stable periodic orbits and regions associated with chaotic behaviors. Bifurcation curves are obtained numerically and analytically. Border collision bifurcations and homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in one-piece are detected.
This paper studies robustness and an application of a one-dimensional window-map based on rotation dynamics. The map with a flat part that is called a window exhibits various superstable periodic orbits (SSPOs) and bifurcations. Using theoretical analysis, we clarify existence regions of the period of the SSPOs in parameter space. We also consider an application to an analog-to-digital converter and clarify output characteristics based on theoretical calculation. Next, we consider the case where the window has a small slope as parameter perturbation and show typical phenomena and robustness. Finally, we discuss bifurcation phenomena and robustness for the window-map based on chaos.
This study mathematically analyzes an interrupted dynamical system (IDS) with a periodic threshold. First, we describe a simple IDS, which is dependent on its own state and a periodic interval, and explain the behavior of the waveform. Then, we define the discrete map (return map) of the system and calculate the bifurcation diagrams. Finally, we focus on the dynamical structure of the return map in the system with a periodic threshold and discuss the stabilizing mechanism, especially its effect in a wide parameter space. The stabilizing effect is verified by the laboratory experiment.
A piece-wise constant (ab. PWC) spiking neuron model (ab. PWN) has a PWC vector field with a state-dependent reset and is designed to reproduce responses of neurons. Based on analysis techniques for discontinuous ODEs, the dynamics of inhibitory responses of the PWN can be reduced into an one-dimensional iterative map analytically. Using the map, it is shown that the PWN can reproduce bifurcation scenarios of inhibitory responses, which are qualitatively similar to those of the Izhikevich model. In addition, a typical bifurcation scenario can be observed in an actual hardware.
An effective algorithm for solving combinatorial optimization problems by using chaotic neurodynamics has already been proposed. Although numerical simulations show that the algorithm is highly efficient, the reason behind its effectiveness has not yet been clarified. In this study, we investigated the searching characteristics of this algorithm for solving combinatorial optimization problems by employing the method of surrogate data, which is frequently used in the field of nonlinear time series analysis. We evaluated how solving abilities depend on bifurcation parameters related to the refractory effects in the chaotic neural networks. Then, we found that the considerable searching ability is decided by refractory effects after neuron firing.
The class-E amplifier is one of the switching amplifiers, which satisfies the class-E switching conditions. It is, however, difficult to determine the values of the passive elements included in the circuit for achieving the class-E switching conditions. The particle swarm optimization algorithm for designing the class-E amplifier is presented in this paper. To reduce the computational cost and consider the nonlinear effects of MOSFET in the amplifier, a circuit simulator which finds the steady-state solution is introduced. Moreover, OpenMP, which is an API for multi-platform shared-memory parallel programming, is used in the developed program to accelerate the proposed procedure. In a design example, the class-E amplifier, the element values of which were obtained by the particle swarm optimization, showed a good performance.
This paper studies digital spike maps that can generate various periodic spike-trains. In order to analyze the dynamics, we consider two measures that are basic to characterize the number of periodic spike-trains and their domain of attraction. As a typical example, we consider a map based on discretized bifurcating neurons. Applying a simple calculation algorithm, we can show that the map can exhibit rich periodic spike-trains as a parameter vary.
This paper presents a deterministic cellular array model of reaction-diffusion systems for parallel generation of pseudo-random binary independent and identically distributed (i.i.d.) sequences. As diffusion systems contain the Brownian particles, the cellular array model contains virtual molecules which move like simple random walkers with their composition changed. Then, the direction which the virtual molecules move in will be i.i.d.. The binary pseudo-random i.i.d. sequences generated with the cellular array were successful in all the randomness tests in NIST FIPS 140-2 and SP 800-22 suites.
In this study, we analyzed the kind of psychological effects that were caused by nonlinear, possibly chaotic vibrations as compared to regular vibrations. For this analysis, we produced a chaotic low-frequency electrical therapy device to generate chaotic vibrations. Using the device, we analyzed the direct effects of chaotic vibrations on the human body. In the experiments, we generated fully chaotic vibrations, intermittent chaotic vibrations, and periodic vibrations. To evaluate the effects of the vibrations on the human body, we used one of the subjective methods; a paired comparison method. We identified a rank-order scale by comparing pairs of two vibrations. The results indicate that complicated vibrations are more effective than periodic vibrations.