Hyperchaotic behaviour of two bi‐directionally coupled Chua's circuits

In this paper, a non‐linear bi‐directional coupling of two Chua's circuits is presented. The coupling is obtained by using polynomial functions that are symmetric with respect to the state variables of the two Chua's circuits. Both a transverse and a tangent system are studied to ensure a global validity of the results in the state space. First, it is shown that the transverse system is an autonomous Chua's circuit, which directly allows the evaluation of the conditions on its chaotic behaviour, i.e. the absence of synchronization between the coupled circuits. Moreover, it is demonstrated that the tangent system is also a Chua's circuit, forced by the transverse system; therefore, its dynamics is ruled by a time‐dependent equation. Thus, the calculus of conditional Lyapunov exponents is necessary in order to exclude antisynchronization along the tangent manifold. The properties of the transverse and tangent systems simplify the study of the coupled Chua's circuits and the determination of the conditions on their hyperchaotic behaviour. In particular, it is shown that hyperchaotic behaviour occurs for proper values of the coupling strength between the two Chua's circuits. Finally, numerical examples are given and discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

Generation of n-double scrolls via cellular neural networks

N-double scrolls are chaotic attractors generated by Chua's circuit when its non-linear resistor characteristic is appropriately modified. They have recently been introduced, simulated and studied analytically by Suykens and Vandewalle. In this paper a new approach to generate n-double scroll attractors is introduced. They have been derived from a connection of three simple generalized cellular neural network cells called a state controlled CNN (SC-CNN). In this way it is established that n-double scroll attractors can be generated using an SC-CNN. The circuit implementation of the introduced system and some experimental results referring to the 2-double scroll attractor are reported.     

On complete stability of linear and quadratic programming neural networks

The paper addresses complete stability (CS) of the important class of neural networks to solve linear and quadratic programming problems introduced by Kennedy and Chua (IEEE Trans. Circuits Syst., 1988; 35 : 554). By CS it is meant that each trajectory converges to a stationary state, i.e. an equilibrium point of the neural network. It is shown that the neural networks in (IEEE Trans. Circuits Syst., 1988; 35 : 554) enjoy the property of CS even in the most general case where there are infinite non‐isolated equilibrium points. This result, which is proved by exploiting a new method to analyse CS (Int. J. Bifurcation Chaos 2001; 11 : 655), extends the stability analysis by Kennedy and Chua (IEEE Trans. Circuits Syst., 1988; 35 : 554) to situations of interest where the optimization problems have infinite solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Some new algebraic criteria for chaos synchronization of Chua's circuits by linear state error feedback control

The research on the sufficient criterion for chaos synchronization of the master–slave Chua's circuits by linear state error feedback control has received much attention and some synchronization criteria for special control matrix were proposed. In this paper, the above synchronization issue is investigated in the situation of general linear state error feedback controller with propagation delay of control signals from the master Chua's circuit. First of all, a master–slave synchronization scheme for Chua's circuits with propagation delay is given and the relevant error system is derived. Using a quadratic Lyapunov function and frequency domain method, three new algebraic synchronization criteria for the synchronization scheme with general control matrix are proven. They are applied to derive the synchronization criteria for simple control matrices. Some examples are given to show the sharpness of these new criteria compared with the known criteria. Copyright © 2006 John Wiley & Sons, Ltd.     

The realization of inverse system for circuits containing nullors with applications in chaos synchronization

A very simple method for synthesizing the inverse system of a non-linear non-autonomous circuit containing nullors is proposed. The main application of the procedure is the synchronization of chaos by the inverse system approach. This is illustrated with two examples: the synchronization of a Duffing circuit and a communication scheme by direct chaotic modulation using Chua's circuit. © 1998 John Wiley & Sons, Ltd.     

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.     

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.     

Rectifier circuits for optimal frequency conversion

It is proved that a circuit consisting of non-linear passive resistances and of any linear invariant passive elements cannot convert power from frequencies ω₁ and ω₂ into power at frequency mω₁ + nω₂ with an efficiency better than 1/(|m| + |n|)². Circuits attaining that efficiency are constructed for all m, n, so that the condition is both necessary and sufficient. For m = μt, n = vt, |μ| + |v| = 2^s (all literals are integers), the optimal circuit consists of a finite number of rectifiers and tuned circuits. For values of m, n that are not of the above form an infinite number of tuned circuits is necessary, but an efficiency better than 93 per cent of the optimum is attainable by simple finite circuits in all cases.     

Devil's staircase route to chaos in a non-linear circuit

A driven second-order negative-resistance oscillator circuit has been observed experimentally to exhibit infinitely many distinct chaotic states in addition to infinitely many subharmonic responses of all orders. Each chaotic state is found to be born out of a devil's staircase whose steps are spaced in accordance with a definite period-adding law. Each devil's staircase emerges at some level of frequency-tuning resolution, where each level is embedded within an outer level, ad infinitum. The global bifurcation structure is self-similar in the sense that upon rescaling, the devil's staircases appear to be clones of each other.     

An integrated model for the study of flicker compensation in electrical networks

This paper presents an integrated model for the simulation of the voltage flicker introduced to a power system due to an arc furnace operation. The model was developed for power system planning purpose in the ATP computational environment. It can be adjusted to the desired operation conditions so as to correctly simulate the furnace operation stages (melting or refining stage) and the expected furnace's degree of flicker severity, for each particular arc furnace and power system combination. Chua's oscillator circuit is used to achieve the chaotic nature behavior of the arc voltage and a Static Var Compensator (SVC) is used for flicker compensation. Modeling and simulation of the full package, including an IEC flickermeter used to evaluate flicker severity, are presented.     

The overdamped double-scroll family. Part I: Piecewise-linear geometry and normal form

This paper is the first part of a two part work that aims to introduce a new family of piecewise-linear differential equations that exhibit chaotic behaviour in numerical simulations and to provide a rigorous mathematical proof of its chaotic nature. The new family presented here has the same eigenvalue distribution as the well-known Lorenz equations in its chaotic regime but has a different symmetry. It is a derivative of the recently investigated double-scroll family of piecewise-linear differential equations.¹ It differs, however, from the double-scroll family in that its members have all real eigenvalues at the origin. In particular, since the associated eigenspace E^s(0) is similar to that of an overdamped linear circuit, we will henceforth refer to this family as the overdamped double-scroll family. We establish Chua's circuit² as a member of this family under appropriate conditions. Preliminary numerical simulations exhibit some interesting phenomena such as period-doubling, periodic windows between chaotic behaviour, and strange attractors of differing geometric structure. Our approach in the rigorous analysis will be that employed for the double-scroll family in Reference 1. We derive a linearly equivalent class of piecewise-linear differential equations which includes the family simulated numerically as a special case. the necessary and sufficient condition for two piecewise-linear vector fields belonging to this new overdamped double-scroll family to be linearly equivalent is that their respective eigenvalues at respective equilibrium points be scalar multiples of each other. If the scalars are all identity, then we have linear conjugacy of the vector fields. an explicit normal form equation, in the sense of global bifurcation, is presented that is parametrized by its own eigenvalues. We find that linearly equivalent vector fields associated with the overdamped double-scroll family exhibit the same global behaviour even though their equivalence is based on the local concept of normalized eigenvalues. In addition, the piecewise-linear differential equations associated with these equivalent vector fields can be quite different from each other.     

High-speed non-linear circuit models for p-n junction diodes

Circuit models for both long-base and short-base p-n junction diodes which are valid for non-linear high-speed and high-frequency operations are presented. the diode model consists of a parallel connection of higher-order dynamic elements and includes the conventional diffusion model as a special case. the new dynamic model can be used for simulating arbitrary p-n junction diode circuits under all operating conditions. In particular, it is capable of simulating realistically the diode's reverse transient behaviour and providing an increasingly accurate approximation to the diffusion equation as the order of the model gets higher. the model is also shown to be capable of reproducing the frequency-dependent small-signal characteristics of p-n junction diodes. The model is based mainly upon the device's physical operating principles. Perhaps the most significant implication of the model is the fact that it illustrates the important roles played by higher-order and dynamic elements in highspeed and high-frequency non-linear device modelling.     

Synchronization of identical neural networks and other systems with an adaptive coupling strength

In this paper, a new scheme to synchronize linearly or nonlinearly coupled identical circuit systems, which include neural networks and other systems, with an adaptive coupling strength is proposed. Unlike other adaptive schemes that synchronize coupled circuit systems to a specified trajectory (or an equilibrium point) of the uncoupled node by adding negative feedbacks adaptively, here the new adaptive scheme for the coupling strength is used to synchronize coupled systems without knowing the final synchronization trajectory. Moreover, the adaptive scheme is applicable when the coupling matrix is unknown or time‐varying. The validity of the new adaptive scheme is also proved rigorously. Finally, several numerical simulations to synchronize coupled neural networks, Chua's circuits and Lorenz systems, are also given to show the effectiveness of the theory. Copyright © 2009 John Wiley & Sons, Ltd.     

‘Lorenz attractor’ from an electrical circuit with uncoupled continuous piecewise-linear resistor

We present a simple piecewise-linear circuit which exhibits a chaotic attractor similar to that observed from the well-known Lorenz equation. Whereas the non-linearities in the Lorenz equation consists of two product terms between two state variables, the non-linearities in our circuit consists of two ‘uncoupled’ 2-terminal continuous non-linear resistors, each characterized by a 2-segment v-i characteristic. Both experiments and computer simulations of this circuit confirm a Lorenz-like strange attractor. the nature of the trajectories associated with this attractor is analysed and explained with the help of the circuit's piecewise-linear-geometry.     

Test circuit of impulse spark‐over voltage with different rates of rise for gas discharge tubes

The impulse spark‐over voltage of the gas discharge tube (GDT) is strongly related to the rate of rise (RoR) of the voltage. At present, due to the convenience of the test instrument, conventional impulse waveforms, such as 1.2/50 and 10/700 μs, are often adopted to conduct such kind of test for GDT. It has been found that these waveforms are able to generate valid waveforms only with limited RoRs and relatively narrow valid voltage ranges. This paper deals with a simple series RC circuit to test the GDT's impulse spark‐over voltage using a ramp voltage, which is approached by an initial portion of a steeply rising voltage impulse whose normal peak value is several times the breakdown voltage of the GDT. The basic circuit theory is described, and the RoR of the output voltage is correlated with the circuit parameters. Moreover, the series RC circuit in this paper is found to be able to deliver a ramp voltage with more consistent RoR than an RLC circuit. The results in this paper provide some suggestions for the revision of relevant standards (e.g., ITU‐T K.12 and IEC 61643‐311). Circuit parameters for different RoRs ranging from 100 V/μs to 1 kV/ns are also established. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

混沌频率调制改善开关电源EMC性能的实验研究

混沌频率调制技术应用于开关电源可以改善开关信号的频谱特性,将原来集中在开关频率各次谐波上的能量分散到各次谐波附近,改善开关电源的EMC性能。文中分别用蔡氏电路、Tent映射、Bernoulli映射产生的离散混沌序列调制Boost电路的开关频率。实验结果表明混沌频率调制可以改善开关电源的EMC性能,同时可以看出不同的离散混沌序列调制后的开关电源EMC性能相差不大。混沌指数增加到一定程度后,开关电源的EMC性能不再有进一步的改善。     

Spatio-temporal co-operative phenomena in CNN arrays composed of chaotic circuits—simulation experiments

This paper presents results of a simulation study of complex dynamic phenomena in arrays composed of interacting chaotic circuits. Such arrays can be thought of as a new paradigm for modelling non-linear phenomena in spatially extended (high-dimensional or infinite-dimensional) systems and active media with potential applications in signal processing. Depending on the connection structure between the cells, the array can show disorganized hyperchaotic behaviour or spatially ordered chaotic waves. Patterns of behaviour depending on the excitation of the array and the connection structure are studied in this paper. Chua circuits are taken as standard chaotic cells.     

Implementing extra element theorem using nullor approach

This letter presents a method of implementing the extra element theorem (EET) on the computer by using the nullor method. The EET involves the calculation of two driving‐point impedances (dpi's), namely the conventional dpi, Z_d, and the null dpi, Z_n. The proposed method is used mainly for calculating Z_n. The method is simplified by representing any given circuit using exclusively RC‐nullor and R‐nullor equivalent circuits, thereby permitting the use of a single (parallel) version of the EET. The proposed method is applied to the linearized boost converter model to derive the fragmented version of the duty‐ratio‐to‐output transfer function. Copyright © 1999 John Wiley & Sons, Ltd.     

Adaptive switching control of uncertain fractional systems: Application to Chua's circuit

Since the introduction of fractional‐order differential equations, there has been much research interest in synthesis and control of oscillatory, periodic, and chaotic fractional‐order dynamical systems. Therefore, in this article, the problem of stabilization and control of nonlinear three‐dimensional perturbed fractional nonlinear systems is considered. The major novelty of this article is handling partially unknown dynamics of nonlinear fractional‐order systems, as well as coping with input saturation along the existence of model variations and high‐frequency sensor noises via just one control input. The method supposes no known knowledge on the upper bounds of the uncertainties and perturbations. It is assumed that the working region of the input saturation function is also unknown. After the introduction of a simple finite‐time stable nonlinear sliding manifold, an adaptive control technique is used to reach the system variables to the sliding surface. Rigorous stability discussions are adopted to prove the convergence of the developed sliding mode controller. The findings of this research are illustrated using providing computer simulations for the control problem of the chaotic unified system and the fractional Chua's circuit model.     

A simple dynamic circuit model for mos structure

A new simple circuit model for the MOS (metal-oxide-semiconductor) structure is presented. the model consists of three elements, namely, a linear capacitor, a non-linear capacitor and a C-dynamic element. Each component bears a simple relationship to the physical operating mechanism inside the MOS structure. the model can be used for simulating arbitrary MOS structure circuits under all operating conditions. In particular, it is capable of reproducing the structure's frequency-dependent small-signal characteristics. the model is also shown to exhibit many important and interesting dynamic behaviours under forward, reverse and sinusoidal operating modes. The model is based mainly upon the device's physical operating principles. But perhaps the most significant implication of this model is that it is the first ever to use a dynamic element to model the MOS structure from a physical approach.