Chaos in Circuits and Systems

The following sections are included:

• Preface

• Contents

An introduction to the design of chaotic oscillators is presented from an electrical engineering point of view. Oscillators are amplifiers with unstable bias points. The basic design principle behind chaotic oscillators is the connection of two electronic circuits which are not in harmony. A number of configurations which may serve as the physical mechanisms behind chaotic behavior are listed. The behavior of an oscillator is explained by means of eigenvalue studies of the linearized Jacobian of the differential equations for the mathematical model of the oscillator. The basic design principle is demonstrated by means of different simple examples.

In this chapter we describe a design cycle for autonomous chaotic oscillators. From a circuit-design perspective and since the linear circuit theory of design still dominates, we find it more appropriate to use the design techniques that have been developed using this theory to establish this design cycle, particularly for the following reasons:

1. The theory of nonlinear dynamics does not yet provide a set of necessary and sufficient conditions for chaos generation.

2. Geometrical interpretation techniques of nonlinear dynamics (Poincaré sections, one dimensional return maps, recurrence plots, etc.) are analysis-biased tools that cannot be related to the design phase of a chaotic oscillator circuit. As for statistical measures, it is not clear whether the available measures are sufficient to identify different types of chaos or to provide comparison guidelines. It is not even known how to produce a system with a predefined dimension, set of eigen-values, Lyapunov exponents and power spectral density distribution. Mapping transformations from nonlinear dynamic concepts to circuit-design concepts are also lacking.

3. On the application front, and to the best of our knowledge, none of the reported applications for chaotic signals strictly refers to a specific type of chaos associated with certain statistical measures or qualitative dynamics. For a circuit designer, it seems that all types of chaos are equally useful. There is no evidence that the chaotic signal produced by a nonautonomous circuit is 'better' than that produced by an autonomous circuit. There is also no evidence that oscillators with active nonlinear devices produce 'better' chaos than those with passive nonlinear ones. The identification, classification, and comparison of different types of chaos from an application perspective is still lacking.

4. The linear circuit theory of design is well established. There are well-known design and optimization techniques, the advantages and disadvantages of which have been studied in detail. The points of weakness of this theory have been noted and are overcome in practice using CAD tools guided by experienced knowledge with an acceptable margin of trial and error.

Therefore, our proposed design methodology aims primarily to minimize the knowledge of nonlinear dynamic concepts that is required by a circuit designer in order to design a chaotic oscillator in a systematic way.

In this chapter, a new design method for chaotic circuits is introduced. This method is simple and flexible, which consists of two oscillators coupled by two diodes. Three kinds of oscillators are applied to the system. Firstly, the system with the same kind of oscillators is investigated via calculation of the Lyapunov exponents in order to confirm the chaos generation and the relations between the ratio of eigenvalues of two oscillators and chaos generation. Secondly, the system with different kinds of oscillators is investigated by determining the ratio of eigenvalues, to verify whether or not chaotic attractors can be observed.

In this chapter, four coupled chaotic circuits that can generate four-phase quasi-synchronization of chaos are proposed. By tuning the coupling parameter, chaotic wandering over the phase states characterized by the four-phase synchronization occurs. In order to analyze chaotic wandering, dependent variables corresponding to phases of solutions in sub-circuits are introduced. Combining the variables with hysteresis decision of the phase states enables statistical analysis of chaotic wandering.

This article reviews various types of intermittent chaos occurring in phase-locked loop systems. First, we will demonstrate a local in-termittency due to a saddle-node bifurcation, which was discovered and named as Type I intermittency by Pomeau and Manneville. Second, we will demonstrate a global intermittency called "Crisis-Induced Intermittency", caused by a tangency of the stable and the unstable manifolds of two saddle points. Third, we will demonstrate an intermittency observed in comparatively high-dimensional systems; on-off intermittency and a new type of intermittency, which has been discovered by us.

In this chapter, we show that using the phase-lock principle is a very fruitful way for advancing in self-excitation and controlling chaotic oscillations, which are important for novel applications, particularly for communication.

In this chapter, we introduce a four-dimensional chaotic oscillator based on two-port voltage-controlled current sources (named 2PVCCSs). The circuit consists of one-linear 2PVCCS, one binary hysteresis 2PVCCS, and two capacitors. The hysteresis 2PVCCS has two jumps that correspond to the third and fourth states apart from the two capacitor voltages. The circuit exhibits double-screw chaotic attractors, quad-screw chaotic attractors, and related periodic attractors. Since the circuit is piecewise linear, the dynamics can be analyzed using the exact piecewise solution. A basic condition for the existence of attractors is given. A transistorized implementation is also presented with some typical SPICE simulations.

In this chapter an overview is given on a class of chaotic and hyperchaotic circuits which can all be represented in Lur'e form. Starting from the well-known Chua's circuit several generalizations are discussed. By introducing additional breakpoints into the nonlinear characteristic one can generate so-called n-scroll attractors. These attractors consist of multiple scrolls which are located on a line in state space. This can be extended to a family of scroll grid attractors, which consist of scrolls located in a plane or a 3D body. Hyperchaotic versions have been obtained for four-dimensional circuits and by weak coupling of chaotic cells within cellular neural networks. Finally a short overview is given about several synchronization methods which have been developed for Lur'e systems, including master-slave and mutual synchronization, robust H_∞ synchronization, time-delayed systems and impulsive synchronization.

This chapter discusses some new circuitry design of chaos generation in simple three-dimensional autonomous systems. The well-known Chua's circuit will be adopted as the framework for basic design. We aim at designing some simple circuits to realize distinct nonlinearities, including smooth nonlinear functions and piecewise-linear functions of one variable, to purposely create chaos. The main theme of the chapter is three-fold. First, we introduce some techniques for chaos generation using smooth nonlinear functions, where a new family of smooth continuous nonlinear functions that can generate multi-scroll attractors is proposed. Second, we present some simple chaos generating circuits by using a piecewise-quadratic function, x|x|, and a piecewise-linear function, respectively, where attractors with multi-scrolls can be easily obtained by adding break points in the piecewise-linear function. Finally, a circuit design for realizing Chen's chaotic attractor, generated by simple smooth quadratic functions, is suggested. The chapter thus presents several snapshots on simple circuitry design for chaos generation.

In this chapter, a current based VLSI degree-two chaos generator is presented. The generator is based upon two unstable oscillators with feedbacks to themselves. The stability of the system is realized via the use of binary hysteresis. The chaotic nature of the signals is guaranteed by the Li-Yorke theorem through the generation of the period-three return map. The initial conditions of the system are discussed and an approach to change them to the origin is proposed. The simulation results are presented finally.

In this chapter we introduce the concept of Frobenius-Perron operators, that is used to study the stochastic aspects of chaotic dynamical systems. We study basic properties and some existence results on fixed densities of this operator. Ulam's method as well as the direct iteration method are also introduced for the computation of the fixed density which gives an absolutely continuous invariant measure underlying the statistical property of the dynamics. The theory and methods are applied to investigate chaotic behavior of electrical circuits.

This chapter describes mixed analog/digital circuit implementations of a chaotic neuro-computer system. The chaotic neuron model is implemented with a switched-capacitor (SC) integrated circuit technique. The analog SC circuit can handle real numbers electrically in the sense that the state variables of the analog circuits are continuous. Therefore, chaotic dynamics can be faithfully replicated with the SC chaotic neuron circuit. The synaptic connections, on the other hand, are realized with digital circuits to accommodate a vast number of synapses. We propose a memory-based digital synapse circuit architecture that draws upon the table look-up method to achieve rapid calculation of a large number of weighted summations. The first generation chaotic neuro-computer with 16 SC neurons and 256 synapses is reviewed. Finally, a large-scale system with 10000 neurons and 10000² synapses is described.

It is known that symmetric standard CNNs are completely stable [1], i.e., each trajectory converges towards some equilibrium point. However, in any practical neural network realization it is not possible to obtain perfect symmetric neuron interconnections, so that the robustness of complete stability with respect to small perturbations of the nominal symmetric neuron interconnections is an extremely important issue. This chapter is devoted to the investigation of such a fundamental issue. The main result obtained is that symmetry of the interconnection matrix is not sufficient in the general case to ensure robustness of complete stability. More precisely, a number of standard CNN configurations are discussed showing that it is possible to prove the existence of stable limit cycles and more complicated attractors, as close to the symmetry condition as one pleases.

Since neural networks have superior information processing functions, many investigators are modeling biological neurons and neural networks. In this chapter, we describe how to construct a pulse-type hardware neuron model based on the asynchronous chaotic neuron model. In particular, we focus on the cell body, describing its circuit construction and chaos dynamics. In addition, we describe the circuit construction and chaos-mediated transmission characteristics of the axon.

We investigate bifurcations of periodic solutions in model equations of two and three Bonhöffer-van der Pol (BVP) neurons coupled through characteristics of synaptic transmissions with a time delay. Bifurcations of the coupled BVP neurons are compared with bifurcations of synaptically coupled Hodgkin-Huxley neurons. We obtain a parameter set of the BVP system, such that the two systems are qualitatively very similar from a bifurcational point of view. This study provides a base for analysis of synaptically coupled neurons with a large number of coupling strategies.

Power electronics is rich in nonlinear dynamics. Its operation is characterized by cyclic switching of circuit topologies, which gives rise to a variety of nonlinear behavior. This chapter provides an overview of the chaotic dynamics and bifurcation scenarios observed in power electronics circuits, emphasizing the salient features of the circuit operation and the modelling strategies. This chapter covers the modelling approaches, analysis methods, and a classification of the common types of bifurcations observed in power electronics.

This chapter gives an evaluation of the chaotic carrier frequency modulation scheme on the spectral characteristics of switching converters. By incorporating a Chua's circuit into the pulse-width modulator for driving the main switch in the converters, different time-and frequency-domain characteristics at equilibrium point, limit-cycle and chaotic regions are observed. The power spectral density of the input current of the three basic dc/dc converters including buck, boost, and buck-boost converters with the Chua's circuit are studied. By controlling the circuit parameter, gradual transition of the spectral characteristics from periodic frequency modulation scheme to near random frequency modulation scheme are observed. Analysis method for random carrier frequency switching scheme is applied and its validity is confirmed theoretically and experimentally.

In the study of nonlinear phenomena in electronics, experiments are indispensable for the purpose of verifying the analytical findings and simulation results. This chapter begins with a discussion on the roles of experimental measurements and computer simulations in the study of nonlinear systems. A tutorial overview of the commonly used laboratory techniques for studying nonlinear phenomena in electronic circuits is given. Specifically, some techniques for displaying phase portraits, Poincaré sections and bifurcation diagrams on the oscilloscope are discussed.

We analyze a Rayleigh type of oscillator, called the Alpazur oscillator, whose bias circuits are switched periodically. We first present a general computational method for finding a bifurcation point for a nonlinear dynamical system with interrupted characteristics. If this system has a chaotic attractor, we also try to stabilize a target unstable periodic orbit embedded in the attractor. We then apply this general methodology to the Alpazur oscillator and show both numerical and experimental results.

Delayed feedback controllers are an appealing tool for stabilization of periodic orbits in chaotic systems. Despite their conceptual simplicity, specific and reliable design procedures are difficult to obtain, partly also because of their inherent infinite-dimensional structure. This chapter considers the use of finite dimensional linear time invariant controllers for stabilization of periodic solutions in a general class of sinusoidally forced nonlinear systems. For such controllers — which can be interpreted as rational approximations of the delayed ones — we provide a computationally attractive synthesis technique based on Linear Matrix Inequalities (LMIs), by mixing results concerning absolute stability of nonlinear systems and robustness of uncertain linear systems. The resulting controllers prove to be effective for chaos suppression in electronic circuits and systems, as shown by two different application examples.

Taming chaos by weak resonant perturbations is a hot topic in recent years. Vast numerical studies have been done, and it has been clarified that taming chaos is generated in many equations. In this chapter, we explain the mechanism for taming chaos by the use of degeneration technique. Our model studied in this chapter is a forced Rayleigh oscillator with a diode. By weak resonant perturbations, chaos is tamed. Two routes to taming chaos are observed. One route is as follows. A wide band chaotic attractor is suddenly changed to a periodic attractor. The other one is as follows. A wide band chaotic attractor is tamed to a narrow band chaotic attractor, and tamed to a periodic attractor. To analyze these phenomena in detail, we will consider the case where the diode in the circuit operate as a switch. In this case, the Poincaré map is derived rigorously as a one-dimensional mapping. By the analysis of the map, it is clarified theoretically that mechanism of the first taming chaos is a saddle node bifurcation, and that the second one is a inverse version of crisis and a inverse period doubling bifurcation.

In a digital communications system, data is transmitted from one location to another by mapping bit sequences to symbols, and symbols to sample functions of analog waveforms. The analog waveforms pass through a band-limited (possibly time-varying) analog channel, where these signals are distorted and noise is added. The analog sample functions sent through the channel are weighted sums of basis functions. In a typical conventional system, the basis functions are sinusoids; in a chaotic communications system, the sample functions are segments of chaotic waveforms.

At the receiver, the symbols may be recovered by means of coherent detection, where all possible sample functions are known, or by noncoherent detection, where one or more characteristics of the transmitted sample functions are determined by the demodulator. In a coherent receiver, synchronization is the most commonly used technique for recovering the sample functions from the received waveform. These sample functions are then used as reference signals for a correlator.

Synchronization-based coherent receivers have advantages over noncoherent receivers in terms of bandwidth efficiency (in narrow-band systems), data rate (in chaotic systems), and noise performance (in both).

These advantages are lost if carrier synchronization cannot be maintained, for example, under poor propagation conditions. Under these circumstances, communication without synchronization may be preferable.

This chapter shows in a tutorial manner how the theory of conventional telecommunications systems can be applied to chaotic modulation schemes. Examples are given of chaotic communications schemes with and without synchronization. The use of chaotic basis functions may give a rise to an estimation problem. A solution to this problem is proposed and the performance of correlator-based systems is evaluated in the context of noisy, band-limited channels. In addition, the multipath performance of differential chaos shift keying, the best chaotic modulation scheme known to date, is discussed.

Nonlinear and possibly chaotic dynamical systems are a source of a very wide range of signals that could be used for solving a variety of signal processing tasks. Given a time series measured (or generated) from a known or an unknown dynamical system we address a series of problems, such as section-wise approximation of the measured signal by pieces of trajectories from a chosen nonlinear dynamical system (model); signal restoration when the measured signal has been corrupted e.g. by quantization; signal coding and compression. All these problems can be approached via estimating the initial conditions for a dynamical system under consideration (generator of approximating waveforms) giving rise to an orbit which is optimal in the sense of distance from the measured (or prescribed) transient output waveform.

In this chapter we consider the problem of state space reconstruction given noisy signals from known chaotic systems. This problem is equivalent to chaos synchronization and is solved with the help of tools from optimal control theory, namely the Kalman filter. The presented approach is introduced for chaotic Lur'e systems to replace the widely used error feedback synchronization. This is in fact an extension of the previously used methods and allows to take into account knowledge about observation noise on the strange attractor to be filtered. We show that the filtering performance is superior to that of error feedback synchronization, in particular if the noise level is non-negligible. Since most of the chaotic electronic circuits are indeed of Lur'e type, the proposed method is suitable for filtering and reconstructing the state of such circuits as used for chaos applications in signal processing or telecommunications.

In this chapter, we consider the problem of identifying an unknown parametrized family of chaotic dynamical systems from a variety of its time series data with a change in the bifurcation parameters. In an experimental situation, in which no a priori analytical knowledge of the dynamical systems is available, we present an algorithm for estimating the underlying bifurcation parameters of the chaotic time series. First, we construct a "qualitatively similar" parametrized family of nonlinear predictors from the sets of chaotic time series. These chaotic time series are then characterized in terms of the "qualitatively similar" bifurcation parameters of the nonlinear predictors. Numerical experiments using the Rössler equations show the efficiency of the algorithm.

This chapter is devoted to a study of secure chaotic communication, particularly for digital chaotic spreading spectrum communication systems. A schematic of an existing spreading spectrum communication system (SSCS) is shown in Fig. 26.1. In this system, the spreading/dispreading module may work in either DS (direct sequence) or FH (frequency hopping) mode. Our study on chaotic SSCS (CSSCS) is in a broad sense: whenever a chaotic PN (pseudo-noise) sequence (shown by the dotted line in Fig. 26.1) plays a role in any SSCS module (shown by the solid line in Fig. 26.1), it is referred to as a CSSCS.

Based on a simple circuit model of a tunneling phase logic (TPL) element that is driven by a sinusoidal voltage source and biased by a DC voltage source, we present simulations of operations in cellular nonlinear networks (CNN) that could potentially be used to perform general computations in 2D arrays of simple, locally connected nanoscale devices. Some examples are presented to demonstrate the image computation capability of TPL-CNN. In particular, we use a simple 2D TPL-CNN structure to perform edge detection, image enhancement and image segmentation. Some cellular automata (CA)-like behaviors of our 2D TPL-CNN are also presented.

This chapter investigates direct methods for evaluating the bifurcation parameter values of periodic solutions in nonlinear systems. The method can numerically calculate parameter values with high convergence rate and accuracy, at which local bifurcations; tangent, period-doubling, Neimark-Sacker and pitchfork bifurcations are occurred. The method uses some basic properties of the characteristic equation for a fixed point of the corresponding Poincaré map. The location and parameter value of the fixed point are used as independent variables in the computation that employs quadratically convergency in Newton method. In addition, in the case of Neimark-Sacker bifurcation, the argument of the complex conjugate multiplier or the other parameter is used as the independent variable. This direct method utilize accurate solution of variational equations with numerical integration, thus implementation of algorithm becomes very simple. Some numerical examples are given.

The objective of this chapter is to provide an introduction to the theory of chaos. One-dimensional maps are used as a vehicle to convey the concepts required for studying chaos, firstly because fewer concepts are involved compared to higher dimensional maps and differential equations and secondly because of the fact that onedimensional maps and their time evolutions may easily be visualized, which helps to grasp ideas involved more rapidly. For generalizations of these results to, and additional results on higher dimensional mappings and differential equations, the reader is referred to the bibliography.

All propositions and theorems are presented without proofs. For the proofs of most of the propositions and theorems refer to [1].

Index