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Mixed-mode oscillations (MMOs) are phenomena observed in a number of dynamic settings, including electrical circuits and chemical systems. Mixed-mode oscillation-incrementing bifurcations (MMOIBs) are among the most complex MMO bifurcations observed in the large group of MMO-generating dynamics; however, only a few theoretical analyses of the mechanism causing MMOIBs have been performed to date. In this study, we use a degenerate technique to analyze MMOIBs generated in a Bonhoeffer–van der Pol oscillator with a diode under weak periodic perturbation. We consider the idealized case in which the diode operates as an ideal switch; in this case, the governing equation of the oscillator is a piecewise smooth constraint equation and the Poincare return map is one-dimensional, and we find that MMOIBs occur in a manner similar to period-adding bifurcations generated by the circle map. Our numerical results suggest that the universal constant converges to 1.0 and our experimental results demonstrate that MMOIBs can occur successively many times. Our one-dimensional Poincare return map clearly answers the fundamental question of why MMOs are related to Farey sequences even though each MMO-generating region in the parameter space is terminated by chaos.

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.

An extensive bifurcation analysis of partial and complete synchronizations of three-frequency quasi-periodic oscillations generated in an electric circuit is presented. Our model uses two-coupled hysteresis oscillators and a rectangular wave forcing term. The governing equation of the circuit is represented by a piecewise-constant dynamics generating a three-dimensional torus. The Lyapunov exponents are precisely calculated using explicit solutions without numerically solving any implicit equation. By analyzing this extremely simple circuit, we clearly demonstrate that it generates an extremely complex bifurcation structure called Arnol’d resonance web. Inevitably, chaos is observed in the neighborhood of Chenciner bubbles around which regions generating three-dimensional tori emanate. Furthermore, the numerical results are experimentally verified.

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.

We investigate the bifurcation phenomena of chaotic attractors observed in electric circuits with piecewise-smooth characteristics. First, we present a circuit model whose switching action depends on its own state and on the clock interval. Next, we explain the behavior of the waveform. Following this, we sample the waveform at every clock period to define the return map, which is vital for a detailed understanding of the circuit dynamics. Finally, bifurcation phenomena of chaotic attractors are classified into four cases with a focus on the invariant interval. In particular, we discuss the characteristics of each bifurcation phenomenon, and then clarify the bifurcation structure of the chaotic attractor. Moreover, some of the numerical results are verified experimentally. © 2014 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

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.

This paper addresses the first experimental demonstration for the nonlinear dynamics in a simple PWM-1 controlled interrupted electric circuit with one dimensional discrete map. First, we show the circuit model and explain its dynamics. Then, the discrete map is mathematically defined for the rigorous analysis. Finally, we show the laboratory experiment and discuss about the circuit's fundamental characteristics.

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.

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.

Switched dynamical systems are known to exhibit border collision, in which a particular operation is terminated and a new operation is assumed as one or more parameters are varied. In this brief, we report a subtle relation between border collision and saddle-node bifurcation in such systems. Our main finding is that the border collision and the saddle-node bifurcation are actually linked together by unstable solutions which have been generated from the same saddle-node bifurcation. Since unstable solutions are not observable directly, such a subtle connection has not been known. This also explains why border collision manifests itself as a "jump" from an original stable operation to a new stable operation. Furthermore, as the saddle-node bifurcation and the border collision merge tangentially, the connection shortens and eventually vanishes, resulting in an apparently continuous transition at border collision in lieu of a "jump." In this brief, we describe an effective method to track solutions regardless of their stability, allowing the subtle phenomenon to be uncovered.

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.

We propose a method for generating a chaotic orbit by using destabilizing control for a stable equilibrium or a stable limit cycle in given differential equation. The controller is designed with the pole assignment technique, which is applied in many linear control systems. In the case of an equilibrium point, the poles of the characteristic equation are located in the right half plane. Control is started when the orbit flows into the point, and is activated during an appropriate interval. Then the orbit is repelled from the point and becomes chaotic. In the case of a limit cycle, the poles of the characteristic equation of the Poincare mapping for a fixed point are assigned as unstable poles. As illustrative examples, a stable equilibrium point of a gradient system, a stable limit cycle of the van der Pol equation, and the extended BVP equation are destabilized and chaotic attractors are obtained. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt. 3, 80(11): 73–81, 1997

In this paper, we analyze stability of the fast-scale and slow-scale dynamics in an interrupted electric circuit. First, we show the full-bridge inverter, as the practical example, in which fast-scale and slow-scale bifurcations are observed. Next, we show our original circuit model and explain its dynamics. Using the sampled data model, we calculate stability of the fast-scale and slow-scale dynamics. Finally, we mathematically show that the period-doubling bifurcation, which occurs in the fast-scale dynamics, does not directly affect the stability of the slow-scale dynamics.

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 discuss, with focus on the border-collision bifurcation, the relationship between the fast-scale bifurcation and the slow-scale bifurcation in an interrupted electric circuit. First, we show the circuit model and explain its dynamics. Then, we define the discrete map in the fast-scale dynamics and the slow-scale dynamics, respectively. Using the discrete map, we derive the 1-parameter bifurcation diagram in the slow-scale dynamics. Finally, we discuss how the border-collision bifurcation in the fast-scale dynamics affects the slow-scale bifurcation.

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, we propose a numerical method to calculate the grazing bifurcation points in an impact oscillator with periodic boundaries. First, we illustrate the n-dimensional autonomous impact oscillator with the moving boundaries. The boundaries consist of the scalar function involving the periodic function. When the trajectory hits one boundary, the solution jumps to the other immediately. Next, we construct the composite Poincare map and show its derivatives. Using the derivatives, we calculate the location of the periodic point, bifurcation parameter and time that elapses before the trajectory hitting the boundary. Finally, we apply the method to a Rayleigh-type oscillator with the sinusoidal boundaries to confirm its validity.

Impact oscillators appear in various fields such as nervous system, ecological system, and mechanical system. These systems have a characteristic property that the dynamics discontinuously behaves due to jumps at hitting borders in the state space. In general, it is difficult to obtain analytical solutions in this class. Thus a numerical method is indispensable for the bifurcation analysis in the impact oscillators; however, unfortunately, it has not been established. Therefore, we proposed a numerical method for the bifurcation analyses in the impact oscillator with a fixed border and applied the proposed method to the Rayleigh-type oscillator.

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.

This paper proposes switch synchronizing delayed feedback control (abbr. SSDFC) in the piecewise linear systems. This control method weeds out the unsuitable control input from the different piecewise continuous functions. As an illustrated example, we then apply this general methodology to a simple circuit containing state-period dependent switch. The unstable periodic orbit is stabilized effectively in the computer simulations. Finally, the performance of SSDFC is compared with delayed feedback control (DFC).

In this paper, we consider a continuous system corresponding to a piecewise-smooth system which consists of four regions. First, we propose a simple circuit whose threshold changes periodically, and derive a return map explicitly. We also consider the fundamental bifurcation phenomena by using one and two parameter bifurcation diagrams. Some theoretical results are verified by the laboratory experiments.

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.

This paper considers a system interrupted by own state and a periodic interval. We know this system has prospects of occurrence of border-collision bifurcation. To analyze properties of the dynamics, we derive a one-dimensional map explicitly. We show some theorems and the existence of regions of periodic solution within two-parameter space. 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.