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An extended Bonhoeffer–van der Pol (BVP) oscillator is a circuit that is naturally extended to a three-variable system from a two-variable BVP oscillator. A BVP oscillator is known to exhibit a canard explosion, and the extended BVP oscillator generates mixed-mode oscillations (MMOs). In this work, we considered a case study where the nonlinear conductor in the extended BVP oscillator includes an idealized diode. The idealized case corresponds to a degenerate case where one of the parameters tends to infinity, and circuit dynamics are represented using a constrained equation, and at the expense of the model’s naturalness, i.e., in a case in which the solutions of the dynamics are defined only forward in time, the Poincare return maps are constructed as one-dimensional (1D). Using these 1D return maps, we explain various phenomena, such as simple MMOs and MMO-incrementing bifurcations. In this oscillator, there exists a small amplitude oscillation, which emerges as a consequence of supercritical Hopf bifurcation, and there exists large relaxation oscillation which appears via canard explosion by changing the bifurcation parameter. Between these small and large amplitude oscillations, the MMO bifurcations exhibit asymmetric Farey trees. Furthermore, these theoretical results were verified using laboratory measurements and experiments.

Nested-layer particle swarm optimization (NLPSO) detects bifurcation parameters in discrete-time dynamical systems. Previous studies have proven the effectiveness of NLPSO for period-doubling bifurcations, but not for other bifurcation phenomena. This paper demonstrates that NLPSO can effectively detect saddle-node bifurcations. Problems in detecting saddle-node bifurcation parameters by conventional NLPSO are clarified, and are solved by imposing a simple condition on the NLPSO objective function. Under this conditional objective function, the NLPSO accurately detected both saddle-node and period-doubling bifurcation parameters regardless of their stability, without requiring careful initialization, exact calculations or Lyapunov exponents.

The bouncing ball system is a simple mechanical collision system that has been extensively studied for several decades. In this study, we investigate the bouncing ball’s dynamics both numerically and experimentally. We implement the system using a table tennis ball and paddle vibrated by a shaker. We focus on the relationship between the ball’s maximum bounce height in the long time interval and the paddle’s vibration frequency, observing several stepwise height changes for frequencies of 25–50 Hz, noting this significant characteristic in both our experiments and numerical simulations. We concentrate on this paddle frequency interval because the phenomenon is easy to handle in numerical simulations. Because the observed phenomenon has a simple order, it can be universal and appear in a large class of collision dynamics. Possibly, some researchers have investigated the bouncing ball system in which the nonsmooth maximum bounce height changes occur. However, they may have failed to notice the changes because the maximal height of the ball was not considered.

In the previous study, a method to control chaos for switched dynamical systems with constant threshold value has been proposed. In this paper, we extend this method to the systems including a periodically moving threshold. The main control scheme is based on the pole placement, then a small control perturbation added to the moving threshold value can stabilize an unstable periodic orbit embedded within a chaotic attractor. For an arbitrary periodic function of the threshold movement, we mathematically derive the variational equations, the state feedback parameters, and a control gain by composing a suitable Poincare map. As examples, we illustrate control implementations for systems with thresholds whose movement waveforms are sinusoidal and sawtooth-shape, and unstable one and two periodic orbits in each circuit are stabilized in numerical and circuit experiments. In these experiments, we confirm enough convergence of the control input.

DC-DC switching converters which are frequently treated as hybrid dynamical systems exhibit complex behavior due to nonlinear and interrupt characteristics. For synchronous buck-converters, we propose a method to control chaotic behavior by pulse-frequency modulation. An input voltage, a duty ratio of PWMs, and so on, affect to the regulation characteristics of converters directly, but a frequency of PWMs is determined by the frequency characteristics of the converter and is set as a fixed value. The proposed chaos-control method suppresses chaotic responses by slightly perturbing the pulse frequency alone, therefore our method can stabilize unstable periodic orbits without influence on the voltage regulation scheme. To simplify the feedback controller, the condition of dimension reduction for the controlling gain vector is derived. The proposed controller acheves the stabilization without a current sensor. Numerical simulation and circuit implementation demonstrate the validity of this method.

This paper proposes a method for calculation of local bifurcation points in discrete-time dynamical systems with piecewise nonlinear characteristics PNDDS. First, an -dimensional PNDDS, which has two piecewise nonlinear maps, is shown and its variational equation is derived. Next, a calculation method for the local bifurcation points that utilizes the conditional equation for the periodic solution and the characteristic equation is proposed. It is essential to calculate the derivatives of the map with an initial value and with a bifurcation parameter to obtain the bifurcation points continuously in the parameter space. The above calculation process is a key component of the proposed method, and is explained in detail. Finally, we apply the proposed method to a two-dimensional PNDDS and calculate the local bifurcation points in order to confirm the validity of the proposed method.

SUMMARY Bifurcation phenomena in a composite dynamic system are studied in order to understand the basic properties of systems. Recently, it has been reported that unavoidable non-ideal switching, e.g., spike noise or a time delay, occurs due to switching and seriously affects the behavior of the trajectory in a composite dynamical system operating in high-frequency switching ranges. We have analyzed the basic properties of a simple one-dimensional composite dynamical system with nonideal switching in order to understand the essence of the dynamical effects of nonideal switching. In the engineering field, there are many two- or more dimensional systems. Naturally, nonideal switching can occur in two- or more dimensional systems. However, no paper analyzes the effect of nonideal switching in such systems. In this paper, we study the basic properties of a two-dimensional composite dynamical system with spike noise. First, we describe a model of a two-dimensional composite dynamical system. Next, the behavior of the waveforms in a system with ideal switching and a system with spike noise is shown. Then, we sample the data of the waveforms in every period for the external force and define a Poincare map. Finally, using the Poincare map, we derive two-parameter bifurcation diagrams and discuss the basic properties of a system with spike noise.

In this study, we analyze the characteristics of an interrupted electric circuit. In particular, we focus on a special situation where the switching action of the circuit is delayed because of a time lag in the response to the switching signal. This situation is observed in switching circuits driven by a high-frequency switching signal. However, the fundamental characteristics of this type of circuit have not yet been clarified. To address this shortfall, we assume that a time lag of the response to the switching signal occurs in simple interrupted electric circuits, and investigate how this time lag affects circuit characteristics. First, we show the model of a circuit whose switching action is the same as that of a current-mode-controlled dc/dc converter. Here by using logic circuits, we impose an artificial time lag on the response to the switching signal. Next, we define a sampled data model (i.e., a return map) that we analyze in detail. Based on the return map, we derive one- and two-parameter bifurcation diagrams. Finally, we compare the bifurcation diagrams constructed with time lag to those constructed without time lag. The results clearly show that time lag is responsible for a new structure in the return map that does not occur in circuits with ideal switching. This new return map structure is a key to understanding the essential characteristics of circuits with time lag. Furthermore, the mathematical results are verified experimentally.

In this paper, we analyze a simple PWM-1 controlled interrupted electric circuit in order to essentially understand the circuit fundamental characteristics. First, we explain the circuit dynamics, and then we define the return map by using the exact solution. Next, we focus on the existence region of the solution invariant interval and bifurcation phenomena in the circuit. In particular, we find the circuit has three types of the invariant interval depending on the parameter. We also clarify that the period-doubling bifurcation and the border-collision bifurcation effect in the existence region of the periodic solution in a wide parameter plane. Finally, the mathematical results are verified by the laboratory experiment. Copyright © 2012 John Wiley & Sons, Ltd.

High-frequency ripple (spike noise) effects in the qualitative properties of DC/DC converter circuits. This study investigates the bifurcation structure of a chaotic attractor in a switched dynamical system with spike noise. First, we introduce the system dynamics and derive the associated Poincare map. Next, we show the bifurcation structure of the chaotic attractor in a system with spike noise. Finally, we investigate the dynamical effect of spike noise in the existence region of the chaotic attractor compare with that of a chaotic attractor in a system with ideal switching. The results suggest that spike noise enlarges an invariant set and generates a new bifurcation structure of the chaotic attractor.

This paper introduces the complete bifurcation analysis in a PWM current-controlled H-Bridge inverter in a wide parameter space. First, we briefly explain the behavior of the waveform in the circuit in terms of the switched dynamical system. Then, the consecutive waveform during the duration of the clock interval is exactly discretized, and the return map is defined for the rigorous analysis. Using the map, we derive the one- and two-dimensional bifurcation diagrams, and discuss the specific property of each bifurcation phenomena in the circuit.

Switching non-ideality and its effects have been reported in DC-DC converters. In this paper, we examine the qualitative property of an interrupted electric circuit with spike noise. First, we show the circuit model that have the switch interrupted by its own state and a periodic interval. Here, we artificially add spike noise via every switching action. Then, we explain its dynamics and derive the Poincare map for the rigorous analysis in a circuit with ideal switching and a circuit with spike noise, respectively. Finally, we discuss the dynamical effects of spike noise from experimental and analytical viewpoints based on the Poincare map and bifurcation diagrams. As a result, some dynamical effects of spike noise are clarified in terms of the invariant set, bifurcations, and existence regions of coexisting attractor. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper, we consider a numerical method for the bifurcation analysis method of nonlinear piecewise-smooth systems . While linear piecewise-smooth systems can be analyzed rigorously, nonlinear piecewise-smooth systems cannot lead to an analytical solution. Derived from both continuous and discrete system analysis approaches, our method uses a Poincare map to transform the results of partial analysis over the continuous components into those issued from a discrete mapping. We then apply conventional methods in order to find critical parameter values and obtain bifurcation diagrams in the parameter space. The numerical procedure is fully described and illustrated by the analysis results of various versions of the Alpazur oscillator.

We have demonstrated that the chaotic circuit with a switching delay is modeled by a return map, and a controller for the suppression of chaos is proposed. A circuit representing a controller stabilizing a period-1 unstable periodic orbit in an interrupted electric circuit with a certain switching delay is also discussed.

In much of the previous study of switched dynamical systems, it has been assumed that switching occurs at a common border between two regions in the same space as the system trajectory crosses the border. However, models arising from this consideration cannot cover systems whose trajectories do not actually "cross" the border. A typical example is the current-mode controlled boost converter whose trajectory is "reflected" at the border. In this paper, we propose a general method to model switched dynamical systems. Also, we suggest an analytical procedure to determine periodic solutions and their stability. The method is developed in terms of solution flows, and no solution has to be explicitly written. Most practical switched dynamical systems can be modeled and analyzed by this method.

This paper shows the stabilization of the unstable periodic orbit of any given piecewise smooth system with linear and/or nonlinear characteristics. By utilizing the periodicity of the switching action, we construct the Poincaremapping including all information of the original system. This mapping offers a first step toward extending a novel technique for controlling chaos based on the appropriate state feedback in piecewise smooth nonlinear systems. We also apply this approach to Rayleigh type oscillator described by the piecewise smooth nonlinear systems. � 2005 Elsevier Ltd. All rights reserved.

SUMMARY In previous works, there are no results about the bifurcation analysis for a piecewise smooth system with non-linear characteristics. The main purpose of this study is to calculate the bifurcation sets for a piecewise smooth system with non-linear characteristics. Werst propose a new method to track the bifurcation sets in the system. This method derives the composite discrete mapping, Poincare mapping. As a result, it is possible to obtain the local bifurcation values in the parameter plane. As an illustrated example, we then apply this general methodology to the Rayleigh-type oscillator containing a state- period-dependent switch. In the circuit, we cannd many subharmonic bifurcation sets including global bifurcations. We also show the bifurcation sets for the border-collision bifurcations. Some theoretical results are veried by laboratory experiments. Copyright ? 2005 John Wiley & Sons, Ltd.

This letter presents an experimental confirmation of controlling the chaotic behavior of a target unstable periodic orbit when the periodically switched nonlinear circuit has a chaotic attractor. The pole assignment for the corresponding discrete system derived from such a nonautonomous system via Poincaré mapping works effectively, and the control unit is easily realized by the window comparator, sample-hold circuits, and so on.

Bonhöffer–van der Pol(BVP) oscillator is a classic model exhibiting typical nonlinear phenomena in the planar autonomous system. This paper gives an analysis of equilibria, periodic solutions, strange attractors of two BVP oscillators coupled by a resister. When an oscillator is fixed its parameter values in nonoscillatory region and the others in oscillatory region, create the double scroll attractor due to the coupling. Bifurcation diagrams are obtained numerically from the mathematical model and chaotic parameter regions are clarified. We also confirm the existence of period-doubling cascades and chaotic attractors in the experimental laboratory.

In this paper, we propose a general method for controlling chaos in a nonlinear dynamical system containing a state-dependent switch. The pole assignment for the corresponding discrete system derived from such a nonsmooth system via Poincaré mapping works effectively. As an illustrative example, we consider controlling the chaos in the Rayleigh-type oscillator with a state-dependent switch, which is changed by the hysteresis comparator. The unstable one- and two-periodic orbits in the chaotic attractor are stabilized in both numerical and experimental simulations.

This paper proposes a bifurcation point detection strategy based on nested layer particle swarm optimization (NLPSO). The NLPSO is performed by two particle swarm optimization (PSO) algorithms with a nesting structure. The proposed method is tested in detection experiments of period doubling bifurcation points in discrete-time dynamical systems. The proposed method directly detects the parameters of period doubling bifurcation regardless of the stability of the periodic point, but require no careful initialization, exact calculation or Lyapunov exponents. Moreover, the proposed method is an effective detection technique in terms of accuracy, robustness usability, and convergence speed.

In this paper, we propose a stability analysis method for the period-1 solution in two-mass impact oscillators. First, we describe a dynamical model and its solution. Next, we define the Poincare map and then we derive derivative of the Poincare map. In particular, we explain the elements of the Jacobian matrix to perform the stability analysis numerically. Finally, we apply this method to a simple two-mass impact oscillator and confirm its validity.

The Nagumo–Sato model is a simple mathematical expression of a single neuron, and it is categorized as a discrete-time hybrid dynamical system. To compute bifurcation sets in such a discrete-time hybrid dynamical system accurately, conditions for periodic solutions and bifurcations are formulated herewith as a boundary value problem, and Newton’s method is implemented to solve that problem. As the results of the analysis, the following properties are obtained: border-collision bifurcations play a dominant role in dynamical behavior of the model; chaotic regions are distinguished by tangent bifurcations; and multistable attractors are observed in its coupled system. We demonstrate several bifurcation diagrams and corresponding topological properties of periodic solutions.

In this paper, we propose a search algorithm of bifurcation point in an impact oscillator with periodic threshold. The algorithm based on the Poincare map approach. In particular, we target the periodic threshold as the bifurcation analysis parameter. First, we define the composite Poincare map of the two-dimensional impact oscillator. Next, we show the derivative of the Poincare map with a parameter of the threshold. Finally, we apply the algorithm for a rigid overhead wire-pantograph system. The validity of the algorithm will be verified by the analyses results.

In this paper, the rigorous analysis is presented for the bifurcation phenomena of a hybrid system containing time delay. First, we introduce a simple system and its behavior of the orbit, and derive a return map explicitly. The fundamental bifurcation phenomenon is analyzed by using one and two parameter bifurcation diagrams. Even in a simple system containing delay time, we discover the two stable orbits coexist, m-piece chaotic attractor, and so on. Finally, we consider the fundamental differences between ideal and time delay system.

In this paper, we investigate the bifurcation phenomena in the nonlinear dynamical system switched by a threshold of the state or a periodic interrupt. First, we propose a method to trace the bifurcation sets for above system. We derive the composite discrete mapping as Poincare mapping. As a result, it is possible to obtain the local bifurcation values in the parameter plane. We also propose an efficient analyzing method for border-collision bifurcations. As an illustrated example, we investigate the behavior of the Rayleigh-type oscillator switched by a threshold of the state or a periodic interrupt. In this system, we can find many subharmonic bifurcation sets including global bifurcations and border collision. Some theoretical results are verified by laboratory experiments.

We try to stabilize unstable periodic orbits embedded in a given chaotic hybrid dynamical system by a perturbation of a threshold value. In conventional chaos control methods, a control input is designed by state-feedback, which is proportional to the difference between the target orbit and the current state, and it is applied to a specific system parameter or the state as a small perturbation. During a transition state, the control system consumes a certain control energy given by the integration of such perturbations. In our method, we change the threshold value dynamically to control the chaotic orbit. Unlike the OGY method and the delayed feedback control, no actual control input is added into the system. The state-feedback is utilized only to determine the dynamic threshold value, thus the orbit starting from the current threshold value reaches the next controlled threshold value without any control energy. We obtain the variation of the threshold value from the composite Poincaré map, and the controller is designed by the linear feedback theory with this variation. We demonstrate this method in simple hybrid chaotic systems and show its control performances by evaluating basins of attraction.