A multi‐wing butterfly chaotic system and its implementation

It has been a common brief that the principle for generating multi‐wing attractor from a Lorenz‐like chaotic system is to construct an even symmetric function for the system. In this pursuit, research effort on exploring the possibility of using non‐even symmetric function is scant, if any. This paper attempts to reveal the intrinsic relation between such a system function and its generated multi‐wing attractor in a chaotic system. It will be shown that a rather simple asymmetric function applying to the Shimizu–Morioka system can indeed generate a multi‐wing butterfly chaotic attractor. Then, some basic properties, compound structure, bifurcation diagram, Lyapunov exponent spectrum, and dynamical analysis of the new chaotic system are investigated in detail. Finally, an electronic circuit is designed and its experimental results are presented, which well match the numerical simulation results, thereby verifying the feasibility of the new method. Copyright © 2017 John Wiley & Sons, Ltd.     

Circuit implementation and model of a new multi‐scroll chaotic system

In this paper, a multi‐scroll chaotic system from the improved Chua's system is proposed. Moreover, non‐linear dynamics are analyzed including phase‐space trajectories, bifurcation diagrams, Poincaré maps and so on. The most important thing is that we discovered phase‐space trajectories, bifurcation diagrams and Poincaré maps are unified and closely related, which can describe different aspects of the multi‐scroll chaotic system. Furthermore, the corresponding improved module‐based circuits are designed for realizing two to four‐scroll chaotic attractors, and the experimental results are also obtained, which are consistent with the numerical simulations. Copyright © 2012 John Wiley & Sons, Ltd.     

On constructing complex grid multi‐wing hyperchaotic system: Theoretical design and circuit implementation

This paper further investigates some novel methods for generating complex grid multi‐wing hyperchaotic attractors from four‐dimensional (4D) quadratic hyperchaotic systems, based on our previous works. First, a modified double‐wing hyperchaotic Lü system by using non‐uniform variable scaling transformation is obtained, and n‐wing hyperchaotic system equipped with a duality‐symmetric multi‐segment quadratic function is also constructed. Then, by switching control in the z direction, mirror symmetry conversion and rotation transformation, three classes of n × m‐wing hyperchaotic systems are respectively realized. Finally, two types of improved module‐based circuits are designed for generating various grid multi‐wing hyperchaotic attractors. One characteristic of the proposed approaches lies in their generality, which is also suitable for constructing 4D grid multi‐wing hyperchaotic Lorenz and Chen systems. Both numerical simulation and circuit implementation have demonstrated the feasibility and effectiveness of the proposed approaches. Copyright © 2010 John Wiley & Sons, Ltd.     

Design and circuit implementation of a five‐dimensional hyperchaotic system with linear parameter

Based on an existing 4‐dimensional Lorenz hyperchaotic system, a novel 5‐dimensional hyperchaotic system is proposed by an extra linear control coefficient in this paper. Some related characters are analysed comprehensively, including the stability of equilibrium, dissipativity, rotation symmetry, bifurcations, Lyapunov exponent, and Poincare surface of a section of the hyperchaotic system. The 5‐dimensional hyperchaotic system is then implemented and simulated in the circuit simulation software NI Multisim. The results indicate that the proposed hyperchaotic system exhibits hyperchaos, chaos, and quasi‐periodic and periodic dynamic behaviours, which can facilitate and enhance the security of chaos‐based communication.     

A minimum five‐component five‐term single‐nonlinearity chaotic jerk circuit based on a twin‐jerk single‐op‐amp technique

A minimum 5‐component 5‐term single‐nonlinearity chaotic jerk circuit is presented as the first simplest chaotic jerk circuit in a category that a single op‐amp is employed. Such a simplest circuit displays 5 simultaneous advantages of (1) 5 minimum basic electronic components, (2) 5 minimum algebraic terms in a set of 3 coupled first‐order ordinary differential equations (ODEs), (3) a single minimum term of nonlinearity in the ODEs, (4) a simple passive component for nonlinearity, and (5) a single op‐amp. The proposed 5‐term single‐nonlinearity chaotic jerk circuit and a slightly modified version of an existing 6‐term 2‐nonlinearity chaotic jerk circuit form mirrored images of each other. Although both mirrored circuits yield 2 different sets of the ODEs, both sets however can be recast into a pair of twin jerk equations. Both mirrored circuits are therefore algebraically twin 5‐component chaotic jerk circuits, leading to a twin‐jerk single‐op‐amp approach to the proposed minimum chaotic jerk circuit. Two cross verifications of trajectories of both circuits are illustrated through numerical and experimental results. Dynamical properties are also presented.     

MOS‐integrable circuitry for multi‐scroll chaotic grid realization: A SPICE‐assisted proof

A MOS‐integrable circuit realization of the class of Multi‐Scroll Grid attractor using an implementation of nonlinear transconductor is presented. The design can be seen as the MOS‐integrable circuit implementation of modified jerk equations presented in the literature (Int. J. Bifurcat. Chaos 2002; 12 (1):23–41). The proposed design of Multi‐Scroll Grid attractor is adequately supported by SPICE simulation results. Copyright © 2008 John Wiley & Sons, Ltd.     

Generating 2n‐wing attractors from Lorenz‐like systems

In this paper, the existence of 2n‐wing chaotic attractors in a family of Lorenz‐like systems is confirmed by both numerical simulation and circuit realization. By replacing a nonlinear cross‐product or square term in an original Lorenz‐like system with a newly designed multi‐segment quadratic function, multi‐wing attractor can be generated. The main design idea is to increase the number of index‐2 equilibrium points of the system. This approach can not only generate multi‐wing attractors in different Lorenz‐like systems, but can also allow the flexibility in specifying a precise number of wings to be created. Copyright © 2008 John Wiley & Sons, Ltd.     

Excitation and steam‐valving coordinated robust controller design for multi‐machine power systems based on the multi‐index nonlinear robust control approach

This paper proposes a novel multi‐index nonlinear robust control (MNRC) approach for multi‐machine power systems. The MNRC approach combines multi‐index nonlinear control with the control theory. With the multi‐index nonlinear control, which selects the output functions as arithmetic combination of state variables, multiple performance indices of the controlled system can be achieved simultaneously in the nonlinear control framework. The control is able to ensure that the system possess the desired robust performance during disturbance. Then, excitation and steam‐valving coordinated robust controllers are developed based on the MNRC approach for multi‐machine power systems. The effectiveness of the proposed robust controller is evaluated by a six‐machine power system simulation. Simulation results show that the expected dynamic and steady‐state performances of power system can be achieved with the MNRC approach. Meanwhile, it is able to achieve the prescribed system performance despite the presence of disturbances. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

A twin‐star hyperchaotic attractor and its circuit implementation

A very simple 4D system with a twin‐star hyperchaotic attractor is presented in this letter. Computer simulation is given to visualize the attractor, and a simple circuitry is designed for system implementation. Copyright © 2003 John Wiley & Sons, Ltd.     

Sliding mode disturbance observer control based on adaptive synchronization in a class of fractional‐order chaotic systems

In this paper, a fractional‐order Dadras‐Momeni chaotic system in a class of three‐dimensional autonomous differential equations has been considered. Later, a design technique of adaptive sliding mode disturbance‐observer for synchronization of a fractional‐order Dadras‐Momeni chaotic system with time‐varying disturbances is presented. Applying the Lyapunov stability theory, the suggested control technique fulfils that the states of the fractional‐order master and slave chaotic systems are synchronized hastily. While the upper bounds of disturbances are unknown, an adaptive regulation scheme is advised to estimate them. The recommended disturbance‐observer realizes the convergence of the disturbance approximation error to the origin. Finally, simulation results are presented in one example to demonstrate the efficiency of the offered scheme on the fractional‐order Dadras‐Momeni chaotic system in the existence of external disturbances.     

An expression for the voltage response of a current‐excited fractance device based on fractional‐order trigonometric identities

We report a closed‐form expression of the voltage response of a current‐excited fractance device. The derived simple expression is made possible following the introduction of the generalized sine and cosine functions (rmsin_α(t) and cos_α(t)), which are valid on any fractional‐order surface and tend to the normal sin(t) and cos(t) at α = 1 or asymptotically as t→∞. Copyright © 2011 John Wiley & Sons, Ltd.     

Delayed feedback control of three‐synchronous‐generator infinite‐bus system

A delayed feedback method for controlling a three‐generator infinite‐bus system is investigated by computer simulation. Nothing but the output signal from the system is used for control. The time series of the phase angle of generators with a transmission line fault shows irregular motion and step‐out. It is found that the irregular motion of the electric power system is coerced onto stable equilibrium points with six stable manifolds by the same control law as the delayed feedback method. © 2006 Wiley Periodicals, Inc. Electr Eng Jpn, 156(1): 7–12, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20329     

Adaptive filtering‐based multi‐innovation gradient algorithm for input nonlinear systems with autoregressive noise

In this paper, by means of the adaptive filtering technique and the multi‐innovation identification theory, an adaptive filtering‐based multi‐innovation stochastic gradient identification algorithm is derived for Hammerstein nonlinear systems with colored noise. The new adaptive filtering configuration consists of a noise whitening filter and a parameter estimator. The simulation results show that the proposed algorithm has higher parameter estimation accuracies and faster convergence rates than the multi‐innovation stochastic gradient algorithm for the same innovation length. As the innovation length increases, the filtering‐based multi‐innovation stochastic gradient algorithm gives smaller parameter estimation errors than the recursive least squares algorithm.     

Control of three‐synchronous‐generator infinite‐bus system by OGY method

Stability of a power system has been investigated in terms of chaos and bifurcation. In this paper, the OGY (Ott–Grebogi–Yorke) method for controlling chaos of three machines operating onto an infinite‐bus system is investigated by computer simulations. The swing equation with the controlling input u is used. The OGY method is extended to the form in the six‐dimensional space. The eight equilibrium points are obtained. The swing equation is normalized and transformed into a discrete‐time state equation from which the control input is calculated. The time series of the phase angles of generators without the control input show the chaotic irregular motion and the step‐out. The time series of the phase angle of generators with the control inputs by the OGY method show the stable motion. The phase angles are successfully controlled onto the unstable equilibrium points with the three unstable manifolds and the three stable manifolds. © 2005 Wiley Periodicals, Inc. Electr Eng Jpn, 151(2): 32–39, 2005; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20008     

Multi‐scroll chaotic attractors from a generalized time‐delay sampled‐data system

In this paper, the first generalization for time‐delay sampled‐data chaotic system in order to generate multi‐scroll attractor is introduced with its circuit implementation. An efficient delay‐line with binary priority encoding, parallel shifting, and binary decoding is also suggested and implemented to overcome the delay line realization drawback in such systems. The proposed system enhances the complexity of chaotic behavior by means of multi‐scroll feature and exemplifies the simplification of chaotic systems for better realizations. Copyright © 2015 John Wiley & Sons, Ltd.     

Complete bifurcation analysis of a chaotic attractor in an electric circuit with piecewise‐smooth characteristics

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.     

A new hyperchaotic Lorenz‐type system: Generation,analysis, and implementation

A new four‐dimensional continuous‐time autonomous hyperchaotic Lorenz‐type system is introduced and analyzed. This hyperchaotic system is not only visualized by computer simulation but also verified with bifurcation analysis and realized with an electronic circuit. Moreover, explicit formulae for estimating the ultimate bound and positive invariant set of the system are derived by constructing a family of generalized Lyapunov functions. The findings and results of this paper have good potential in control and synchronization of hyperchaos and their engineering applications. Copyright © 2010 John Wiley & Sons, Ltd.     

Analysis and realization of a switched fractional‐order‐capacitor integrator

Using fractional calculus, we analyze a classical switched‐capacitor integrator when a fractional‐order capacitor is employed in the feed‐forward path. We show that using of a fractional‐order capacitor, significantly large time constants can be realized with capacitances in the feedback path much smaller in value when compared with a conventional switched‐capacitor integrator. Simulations and experimental results using a commercial super‐capacitor with fractional‐order characteristics confirmed via impedance spectroscopy are provided. Copyright © 2016 John Wiley & Sons, Ltd.     

Output‐feedback adaptive tracking control of stochastic nonlinear systems using multi‐dimensional Taylor network

In this paper, an adaptive multi‐dimensional Taylor network (MTN) control scheme based on the backstepping and dynamic surface control (DSC) is developed to solve the tracking control problem for the stochastic nonlinear system with immeasurable states. The MTNs are used to approximate the unknown nonlinearities, and then based on the multivariable analog of circle criterion, an observer is first introduced to estimate the immeasurable states. By combining the adaptive backstepping technique and the DSC technique, an adaptive MTN output‐feedback backstepping DSC approach is developed. It is shown that the proposed controller ensures that all signals of the closed‐loop system are remain bounded in probability, and the tracking error converges to an arbitrarily small neighborhood around the origin in the sense of probability. Finally, the effectiveness of the design approach is illustrated by simulation results.     

Reactive power compensation in a multi‐line system under sinusoidal unbalanced conditions

A new generalized reactive power compensation algorithm applicable to a multi‐line system under sinusoidal conditions is reviewed. This method does not require any decomposition of reactive power or non‐active current. It is shown that a realization of the compensator can be obtained by means of a simple connection of reactive elements. Accordingly, it is concluded that to compensate the reactive power is a simple circuit synthesis problem. The algorithm is illustrated by means of examples. Copyright © 2010 John Wiley & Sons, Ltd.