Chua's Circuit: A Paradigm for Chaos

The following sections are included:

• PREFACE

• CONTENTS

• GALLERY OF ATTRACTORS FROM CHUA'S OSCILLATOR

Three main types of strange attractors are described; namely hyperbolic, Lorenz-type and quasiattractors. In addition, a recent family of quasiattractors originating from Chua's circuit is briefly described. In connection with the strange attractors, we stress that models having quasiattractors containing structurally unstable homoclinic Poincaré orbits are “bad”, in the sense of Ref. 24.

This paper presents design considerations for a 2.4 μm CMOS prototype of the Chua's circuit. Our design uses VCCSs, realized using the quasilinear region of the transfer characteristics of a differential pair. The global, nonlinear characteristics of this building block are exploited to realize the Chua's diode. The prototype has been fabricated in a 2.4 μm double poly n-well CMOS technology and occupies 0.35 mm². Power consumption is 1.6 mW for a symmetrical biasing of ±2.5 v. Circuit operation is controlled by changing differential pair bias currents. Measurements show bifurcation towards a double-scroll Chua's attractor by changing a bias current.

In this paper a comparative study of the Lorenz equation and Chua's equation is presented.

In this introductory tutorial paper we show how Chua's circuit can be used to observe chaotic phenomena experimentally. Several aspects of chaotic systems will be demonstrated in Chua's circuit.

In particular, we present a picture book of bifurcation and chaotic phenomena occurring in Chua's circuit. We show a bifurcation sequence where a stable fixed point loses stability and an attracting limit cycle appears as a parameter is varied. This limit cycle then undergoes a period-doubling sequence where the periodicity of the limit cycle is doubled repeatedly until chaos is observed.

Some questions concerning computer simulation of the corresponding differential equations are discussed, and we also demonstrate how the period-doubling sequence in Chua's circuit converges in accordance with the Feigenbaum universal number by calculating the ratios of the parameter change at successive bifurcation points. Finally, diagrams of eigenvalue patterns in the α-β parameter plane are presented.

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua's circuit. We especially emphasize some properties of the confinors of Chua's equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincaré maps which reveal the precise structures of Chua's strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates.

We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters.

Chua's equation seems surprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua's equation and the use of sequences of Taylor's coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua's equation, leads instead to a very accurate analysis of this phase portrait.

Two different bifurcation patterns are experimentally observed in Chua's circuit. They show that antimonotonicity — inevitable reversals of period-doubling sequences, is a typical phenomenon in Chua's circuit.

We present the bifurcation analysis of Chua's circuit equations with a smooth nonlinearity, described by a cubic polynomial. Our study focuses on phenomena that can be observed directly in the numerical simulation of the model, and on phenomena which are revealed by a more elaborate analysis based on continuation techniques and bifurcation theory. We emphasize how a combination of these approaches actually works in practice. We compare the dynamics of Chua's circuit equations with piecewise-linear and with smooth nonlinearity. The dynamics of these two variants are similar, but we also present some differences. We conjecture that this similarity is due to the central role of homoclinicity in this model. We describe different ways in which the type of a homoclinic bifurcation influences the behavior of branches of periodic orbits. We present an overview of codimension–1 bifurcation diagrams for principal periodic orbits near homoclinicity for three-dimensional systems, both in the generic case and in the case of odd symmetry. Most of these diagrams actually occurs in the model. We found several homoclinic bifurcations of codimension–2, related to the so-called resonant conditions. We study one of these bifurcations, a double neutral saddle loop.

Chaos has been widely reported and studied in Chua's circuit family, which is characterized by a 21 parameter family of odd-symmetric piecewise-linear vector fields in R³. In this tutorial paper, we shall prove that, up to a topological equivalence, all the dynamics of this family are subsumed within that of a single circuit: Chua's oscillator; directly derived from Chua's circuit by adding a resistor in series with the inductor. We provide explicit formulas of the parameters of Chua's oscillator leading to a behavior qualitatively identical to that of any system belonging to Chua's circuit family. These formulas are then used to construct, in an almost trivial way, a gallery of (quasiperiodic and strange) attractors belonging to Chua's circuit family. A user-friendly program is available to allow a better understanding of the evolution of the dynamics as a function of the parameters of Chua's oscillator, and to follow the trajectory in the eigenspaces.

We present a picture book of unstable periodic orbits embedded in typical chaotic attractors found in the canonical Chua's circuit. These include spiral Chua's, double-scroll Chua's and double hook attractors. The “skeleton” of unstable periodic orbits is specific for the considered attractor and provides an invariant characterisation of its geometry.

One of the most important tasks in the analysis of a nonlinear system is to determine its global behavior and, in particular, to delineate the domains of attraction for asymptotically stable solutions. Stable manifolds often act as boundary surfaces between such domains in the state space.

In this paper the morphology of boundary surfaces is studied in a single member of Chua's circuit family, although the techniques used apply equally well to many other nonlinear circuits.

Of all the PWL circuits known so far which exhibit two stable states it is typical that their resistor characteristics each have at least three segments. Although bistability cannot be achieved via a 2-segment characteristic in the plane, complicated bistable behavior, including chaotic attractors, can occur locally at the boundary of two linear regions in the 3-D Chua's circuit, i.e. bistability is achieved with a minimal number of segments. By using a 3-segment characteristic, at least five attractors can be generated. Basin structure of the corresponding attractors is examined using numerical simulations.

Period-adding, symmetry-breaking and remerging bifurcation phenomena are observed experimentally and numerically from an extended Chua's circuit.

Dynamical properties of sequential circuits can be investigated by means of switching between the system's attractors, and boundary surfaces play a crucial role in the process of switching. The use of basin delineation in the triggering of multistable circuits is shown.

In this paper, we report numerical observations of the stochastic resonance (SR) phenomenon in a bistable chaotic electronic circuit (namely, Chua's circuit) driven simultaneously by noise and a sinusoidal signal. It is shown that the noise-induced “chaos-chaos” type intermittency is a physical mechanism of the SR-phenomenon in chaotic systems. The resulting amplification of the sinusoidal signal intensity is due to a coherent interaction of three characteristic frequencies of the system. The SR-phenomenon can be controlled by a variation of either the noise intensity or the system parameters in the absence of noise.

Experimental evidence showing the existence of signal amplification via perturbation of periodic and chaotic orbits in an electronic circuit is presented.

A number of recent papers have investigated the feasibility of synchronizing chaotic systems. Experimentally one of the easiest systems to control and synchronize is the electronic circuit. This paper examines synchronization in Chua's Circuit, proven to be the simplest electronic circuit to exhibit chaotic behavior.

In this paper sufficient conditions that each cell in a one-dimensional array of Chua's circuits has an identical and synchronized behavior are given.

Traveling wave fronts are considered for a one-dimensional array of Chua's circuits. This solution is obtained analytically and analyzed for the “primary real bifurcation”. For diffusion coefficients less than some nonzero critical value it has been observed numerically that the traveling fronts fail to propagate. This nonlinear phenomena is similar to that observed from pulse propagation in nerves, and in coupled continuously-stirred tank reactors.

Dynamics of a chain of coupled Chua's circuits is investigated. Existence of spatial disorder is revealed. Dynamics of wave fronts in the limiting case of a continuous medium is studied.

Secure communications via chaotic synchronization is experimentally demonstrated using Chua's circuit. In our experiment the energy lost in the information-bearing signal is approximately 0.6 dBV. The reduction in chaotic signal after the recovery process is between −4 0 and −50 dBV.

In this paper we demonstrate experimentally how Chua's circuits can be used to implement a secure communication system. This system is compared to another system proposed earlier. This system has the advantage of transmitting a spread spectrum signal and is also more secure in the sense that better parameter matching is required in order to recover the signal. Furthermore, there is less of a problem with loss of synchronization.

The transmission of digital signals by means of chaotic synchronization is demonstrated, numerically as well as experimentally, via Chua's circuit.

Low level vision for feature detection, motion analysis, or image segmentation is typically performed in parallel and is computationally intensive. The dynamic nature of scenes and the requirements for real time processing place further demands upon visual sensing. Dynamical systems which mimic the complexity of natural scenes provide an alternative to traditional computer vision approaches. However the design of such systems and the synthesis of complex, nonlinear dynamical systems by the interactions of simpler, low order systems remains a critical problem.

One approach to this problem is to use the relative simplicity of Chua's circuit to provide a convenient model for the dynamics and bifurcation phenomena in more complex systems. In this paper the normal form is derived for Chua's circuit in which the piecewise-linear function is replaced by a cubic nonlinearity. A partial bifurcation analysis of the normal form equations is then used to show how Chua's system can be made to track the motion of low level image features through parameter variations in the bifurcation function.

Nonlinear Dynamics have been very inspiring for musicians, but have rarely been considered specifically for sound synthesis. We discuss here the signals produced by Chua's circuit from an acoustical and musical point of view. We have designed a real-time simulation of Chua's circuit on a digital workstation allowing for easy experimentation with the properties and behaviors of the circuit and of the sounds. A surprisingly rich and novel family of musical sounds has been obtained. The audification of the local properties of the parameter space allows for easy determination of very complex structures which could not be computed analytically and would not be simple to determine by other methods. Finally, we have found that the time-delayed Chua's circuit can model the basic behavior of an interesting class of musical instruments.

The occasional proportional feedback (OPF) control technique has been successful in stabilizing periodic orbits in both periodically driven and autonomous systems undergoing chaotic behavior. By applying this technique to the well-known Chua's circuit, we are able to control a variety of periodic orbits including single-correction, low-period orbits and multiple-correction, high-period orbits. Also, by employing two control circuits, we are able to stabilize orbits that visit both regions of Chua's circuit's double-scroll attractor, applying corrections in each of these regions during a single orbit.

Systems often have limited operating regions at increased drive parameters because of the onset of oscillations and chaos. Chua's circuit is an electronic example of such a system, as it has a nonzero steady-state (d.c.) voltage output before breaking into oscillations and eventually chaos as the drive is increased. Using a new control technique, we are able to keep the system in the steady state in regions for which it would otherwise exhibit oscillations or chaos.

The nonautonomous version of Chua's circuit can exhibit a wide variety of bifurcation routes to chaos. In this paper we explore the possibility of controlling chaos in this circuit by the addition of a second periodic signal for certain parametric choices.

We discuss the problem of directing trajectories on the double-scroll Chua's attractor to a neighbourhood of unstable stationary states in a finite a priori known time.

The unified canonical feedback control strategy developed recently by the present authors for controlling chaotic systems is refined and applied to the well-known Chua's circuit, driving its orbits from the chaotic attractor to its unstable limit cycle. Simple sufficient conditions for the controllability of this particular circuit are established. Simulation results are included to visualize the control process. A circuit implementation of the designed feedback control is realized by adding a linear resistor and an appropriate periodic-signal generator to the original circuit.

This paper considers the control of a polynomial variant of the original Chua's circuit. Both state space techniques and input-output techniques are presented. It is shown that standard control theory approaches can easily accommodate a chaotic system. Furthermore, it is shown that a harmonic balance approach could predict the period doubling phenomenon and onset of the double scroll chaos, as well as providing a control approach.

The paper presents a new approach to the control of chaotic systems for the stabilization of a periodic orbit. The problem formulation requires preserving a number of original system characteristics and making use of a low energy control. The proposed method follows a frequency harmonic balance technique employed in the approximate analysis of complex nonlinear phenomena. The design procedure is applied to the well-known Chua's circuit showing the main characteristics of the approach: it gives reasonably accurate results, partly intuitive in nature, by means of quite simple computations.

We propose a simple method for the synchronization and the control of chaos. A general qualitative description of the method is presented.

This paper is written as a tutorial on how to use a one-dimensional map derived from Chua's circuit to study the circuit's complicated dynamics. While the derivation of this 1-D map is nontrivial, a user-friendly program is presented to help the beginner uncover and witness, without any prior background on chaos, numerous periodic, homoclinic, heteroclinic and chaotic orbits. In keeping with the pedagogical nature of this paper, these bifurcation phenomena will be profusely illustrated with pictures generated from a computer program, along with the exact parameters so that the reader can easily duplicate them. The program is written in the C-language for both PC-486 computers and UNIX workstations.

Bifurcation analysis by means of a 1-D return map obtained experimentally from Chua's circuit is presented. It is experimentally demonstrated that a homoclinic bifurcation associated with the stationary point at the origin of the phase space precedes the birth of the double-scroll Chua's attractor.

A complex fine structure in the topography of regions of different dynamical behavior near the onset of chaos is investigated in a parameter plane of the 1D Chua's map, which describes approximately the dynamics of Chua's circuit. Besides piecewise-smooth Feigenbaum critical lines, the boundary of chaos contains an infinite set of codimension-2 critical points, which may be coded by itineraries on a binary tree. Renormalization group analysis is applied which is a generalization of Feigenbaum's theory for codimension-2 critical points. Multicolor high-resolution maps of the parameter plane show that in regions near critical points having periodic codes, the infinitely intricate topography of the parameter plane reveals a property of self-similarity.

A discrete map can be obtained from Chua's circuit¹ with a discontinuous nonlinearity. Chaotic behavior is observed in the simulation of Chua's circuit with a discontinuous piecewise-linear element. The Poincaré map can be used to study the stability behavior and opens the possibility for analyzing the chaotic behavior by determining the existence of a snap-back repeller.

We analytically derive a one-dimensional map from an ODE which produces a double scroll very similar to the Chua double scroll. Our analysis leads us to suggest a generalization of the Chua circuit to an n-dimensional system of ODEs that we will call the generalized Chua equations. The third order system of ODEs in this class contains the Chua equations as a special case. Parallel to the generalized Chua equations we define the generalized Chua maps. An important feature of these equations and maps is that the source of their nonlinearity is a sigmoid function, and functions very similar in their properties to the sigmoid function. We show that this class of equations contains examples of maps that reproduce the Lorenz and Rössler dynamics. We suggest that a general theory of these equations and maps, and their relationship to one-dimensional maps, is possible.

A benefit of our analysis shows that the dynamics of the maps of Rössler, Chua, and Lorenz maps can be traced to a common set of building blocks, and we conclude that the Chua map is the simplest of the three maps and therefore understanding the complexity in the Chua map provides a foundation for understanding chaos in a large class of n-dimensional equations that includes the maps of Rössler and Lorenz.

Following Brown^1,2 we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of the pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1 − x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

The double-hook family— of three-region, piecewise-linear, continuous vector fields on R³—is a derivative of the double-scroll family that also numerically and experimentally displays complex bifurcation phenomena and chaotic behavior. Salient analytical results culled from a comprehensive formal study of the qualitative dynamics generated by are presented. In particular, the findings cover (1) the relationship between , the double-scroll family, and Chua's circuit; (2) the generic simplification techniques used in the study of the dynamics, which involve coordinate transformations, canonical vector fields, and a normal-form equation; (3) the characteristic Poincaré maps that capture the qualitative behavior exhibited by and that describe the structure of the double-hook attractor; (4) the formal existence of “horseshoe” chaos for a particular member of that employs Shil'nikov theory; and (5) the extension of results to the complementary dual double-hook family. Findings are generously illustrated geometrically and are compared with those for the double-scroll family.

We apply the singular perturbation method to Chua's equations, considered as a slow-fast autonomous dynamical system (S-FADS). An algorithm available for most nonlinear S-FADS generalizes the singular perturbation method, gives the nth-order approximation equations, and allows us to go further with a qualitative study. This algorithm is applied to the Chua's system in a very simple way. Indeed, because this system is integrable, it is possible to compute explicitly the slow and fast dynamics for a better understanding of the nonlinear cases.

A canonical Chua's circuit is investigated as a feedback automatic control system. Frequency-domain criteria of global stability and instability are presented. New version of the generalized Kalman's conjecture is considered.

We present a new type of strange attractors generated by an odd-symmetric three-dimensional vector field with a saddle-focus having two homoclinic orbits at the origin. This type of attractor is intimately related to the double-scroll Chua's attractor. We present the mathematical properties which proved rigorously the chaotic nature of this strange attractor to be different from that of a Lorenz-type attractor or a quasi-attractor.

In particular, we proved that for certain nonempty intervals of parameters, our two-dimensional map has a strange attractor with no stable orbits. Unlike other known attractors, this strange attractor contains not only a Cantor set structure of hyperbolic points typical of horseshoe maps, but also there exist unstable points (i.e. stable in reverse time) belonging to the attractor as well. This implies that the points from the stable manifolds of the hyperbolic points must necessarily attract the unstable points.

We discuss properties of the double-scroll Chua's attractor in comparison with other known attractors; e.g. the Lorenz, Lozi and Hénon attractors. An approach for studying the hyperbolic trajectories in this attractor is proposed. We assume that the double-scroll Chua's attractor has unstable directions and give some confirmation of this conjecture.

This paper is concerned with exploiting the architecture of a single-chip digital signal processor for integrating piecewise-linear ODEs. We show that DSPs can be usefully applied in the study of Chua's circuit family provided that one chooses a multistep integration algorithm which exploits their unique single-instruction multiply-and-accumulate feature.

A special purpose program has been developed to compute the time-domain solutions and the Lyapunov exponents for Chua's circuit family. Advantage has been taken of the piecewise-linear nature of the circuits and the small number of linear regions in order to obtain fast and accurate computations. Examples are given to illustrate the capabilities of the program, in particular, to illustrate the calculation of the Lyapunov exponents.

Experiments with modelling and simulation of Chua's circuit are presented in order to transfer a simple explanation of its behaviour from the wonderful fairy-tale world of mathematicians to the marvellous physical world of electronic engineers. It is demonstrated that the behaviour of the circuit is based on the interaction (superposition) of two different kinds of energy balance: (1) Chaotic behaviour based on the interaction of two unstable DC-states and (2) stable limit cycle behaviour based on the balance between the energy lost in the regions with mainly positive losses and the energy gained in the regions with mainly negative losses. Convergence problems observed in connection with simulation of the ideal piecewise-linear model are solved by means of a smooth continuous model of the nonlinear element based on the ideal operational amplifier model instead of a polynomial approximation. The preliminary results are presented. The NAP2 program made by the late Thomas Rübner-Petersen has been used for the simulations.

This paper includes a method for detecting the maximum possible range of bifurcations based upon the multilevel oscillation technique. An application of the method to Chua's circuit, and new simulation results using the slope of the piecewise-linear function as a bifurcation parameter are presented.

The behavior of transfer and return maps in the intermediate region of Chua's circuit and related systems undergoes a number of changes as the symmetry of the dynamics is broken, i.e. the separating planes are moved away from symmetric positions. We employ the technique of maps induced by the flow of the system and construct the critical curves for the maps in the intermediate region of state space. The influence of a broken symmetry on the critical curves and the flow is discussed in depth. We demonstrate that any breaking of symmetry potentially weakens and eventually destroys the chaos producing mechanisms.

This paper deals with the inductor-free realizations of the four known circuits (see Refs. 1, 2) belonging to Chua's circuit family by using a simple feedback system consisting of a linear passive RC-filter and a nonlinear amplifier. The basic approach and the design principles and the parameter ranges, where equivalent RC-circuits can be realized are given.

Statistical and dynamical properties of the “chaos–chaos” type intermittency in Chua's circuit are studied by numerical simulation methods. It is shown that at the onset of this intermittency phenomenon, the power spectrum of the associated time series has the shape of 1/f noise. The influence of external noise and computer calculation errors on the form of the spectrum and other process characteristics are analyzed. The structure and scaling of the Poincaré map at the transition threshold are also discussed.

Bispectral analysis (which isolates quadratic nonlinear interactions among triads of Fourier components) is used to investigate bifurcations in Chua's circuit. For period-doubled limit cycles, the dominant interactions of the circuit are quadratically nonlinear, and bicoherence spectra isolate the phase coupling between increasing numbers of triads of Fourier components as the nonlinearity of the system is increased. For circuit parameters that result in a chaotic, Rössler-type attractor, bicoherence spectra indicate that quadratic nonlinear interactions are important to the dynamics. For parameters that lead to the double scroll chaotic attractor the bispectrum is zero, suggesting that nonlinear interactions of order higher than quadratic dominate the dynamics. Higher-than-second order spectra (e.g. trispectra) are required to isolate the individual nonlinear interactions for the double scroll.

The Chua circuit can be treated as a black box giving output which can be studied in a number of ways. This allows us to test some novel nonlinear system modehng methods. In particular, we show how to locate important dynamical features such as fixed points and near-heteroclinic cycles with very little effort, and then how to build a more detailed dynamical model to verify these discoveries.

The estimation of Lyapunov exponents from experimental data is demonstrated by means of a chaotic time series from Chua's circuit. The changes in the Lyapunov spectrum upon time reversal, the local thickness of the attractor, and the approximation errors of the locally linear fits are used as diagnostic tools to identify spurious Lyapunov exponents and to monitor the reliability of the results.

Cellular neural networks (CNN) are time-continuous nonlinear dynamical systems. Like in Chua's circuit, the nonlinearity of these networks is defined as a piecewise-linear function. For CNNs with at least three cells chaotic behavior may be possible in certain regions of parameter values. In this paper we report such a chaotic attractor with an autonomous three-cell CNN. It can be shown that the attractor has a structure very similar to the double-scroll Chua's attractor. Through some equivalent transformations this circuit, in three major subspaces of its state space, is shown to belong to Chua's circuit family, although originating from a completely different field.

A chaotic neural network, called chaotic Cellular Neural Network (CNN), is proposed for performing complex information processing tasks. Each cell in the chaotic CNN is a Chua's circuit and connected only to its nearest neighbors. The proposed network of coupled Chua's circuit type cells constitutes a special case of the generalized CNNs introduced recently.¹ Individual cells play the role of an analog microprocessor: producing constant, oscillatory or chaotic steady-state outputs depending on its input, which is the weighted sum of external inputs and the outputs of neighboring cells. The proposed chaotic CNN has complex temporal dynamical behaviours and hence provides a potentially rich mechanism for information processing, specially for nonlinear signal processing.

In this paper we present generalized theorems of the Shil'nikov type for evolution equations in Banach spaces of infinite dimension, which describe the behaviour of subsystems of solutions in a neighborhood of a double homoclinic orbits to the same saddle-focus point.

The preliminary results of experimental investigations of chaotic oscillations from Chua's circuit at high frequencies are reported. The double-scroll Chua's attractor with the usual set of LC-circuit parameters was observed at 100 KHz. A new region in the parameter space of Chua's circuit was discovered where the double-scroll Chua's attractor exists which allows an increase in the working frequency of up to 300 KHz. The highest central frequency f = 7.65 MHz of a chaotic oscillation was obtained for the case of Chua's circuit with a one-mode coaxial resonator. The ergodic torus and resonant tori with different rational Poincaré rotation numbers are observed. The possibility for obtaining from Chua's circuit the spread spectrum chaotic signals was shown.

We study the bifurcations of attractors of a one-dimensional 2-segment piecewise-linear map. We prove that the parameter regions of existence of stable point cycles γ are separated by regions of existence of stable interval cycles Γ containing chaotic everywhere dense trajectories. Moreover, we show that the period-doubling phenomenon for cycles of chaotic intervals is characterized by two universal constants δ and α, whose values are calculated from explicit formulas.

In this paper, we consider an infinite-dimensional extension of Chua's circuit (Fig. 1) obtained by replacing the left portion of the circuit composed of the capacitance C₂ and the inductance L by a lossless transmission line as shown in Fig. 2. As we shall see, if the remaining capacitance C₁ is equal to zero, the dynamics of this so-called timedelayed Chua's circuit can be reduced to that of a scalar nonlinear difference equation. After deriving the corresponding 1-D map, it will be possible to determine without any approximation the analytical equation of the stability boundaries of cycles of every period n. Since the stability region is nonempty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the “period-adding” phenomenon where every two consecutive cycles are separated by a chaotic region.

The following sections are included:

• ADVENTURES IN BIFURCATIONS AND CHAOS