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We study synchronization in an array of coupled nonlinear systems with delay and nonreciprocal time-varying coupling and present synchronization criteria which generalize previous synchronization results. We show that the array synchronizes when the nondelay coupling term is cooperative and large enough. Furthermore, we show that the synchronization criteria are related to several matrix quantities describing the coupling topology. These quantities can be considered as generalizations of the concept of algebraic connectivity to directed graphs. Index Terms--Delay systems, directed graphs, Lyapunov methods, nonlinear systems, synchronization.

We present some new global stability results of neural networks with delay and show that these results generalize recently published stability results. In particular, several different stability conditions in the literature which were proved using different Lyapunov functionals are generalized and unified by proving them using the same Lyapunov functional. We also show that under certain conditions, reversing the directions of the coupling between neurons preserves the global asymptotical stability of the neural network. Index Terms--Asymptotical stability, delay equations, Lyapunov functional, neural networks.

There are, in general, two classes of results regarding the synchronization of chaos in an array of coupled identical chaotic systems. The first class of results relies on Lyapunov's direct method and gives analytical criteria for global or local synchronization. The second class of results relies on linearization around the synchronization manifold and the computation of Lyapunov exponents. The computation of Lyapunov exponents is mainly done via numerical experiments and can only show local synchronization in the neighborhood of the synchronization manifold. On the other hand, Lyapunov's direct method is more rigorous and can give global results. The coupling topology is generally expressed in matrix form and the first class of methods mainly deals with symmetric matrices whereas the second class of methods can work with all diagonalizable matrices. The purpose of this brief is to bridge the gap in the applicability of the two classes of methods by considering the nonsymmetric case for the first class of methods. We derive a synchronization criterion for nonreciprocal coupling related to a numerical quantity that depends on the coupling topology and we present methods for computing this quantity. Index Terms--Chaos, convex programming, coupled oscillators, Lyapunov exponents, Lyapunov functions, nonlinear programming, nonreciprocal coupling, synchronization.

The property of any two-terminal resistive device is characterized by its driving-point (DP) characteristic, and any two-port resistive device with zero input current and a load-independent output voltage is described by its transfer characteristic (TC). Synthesizing a two-port device with a prescribed transfer characteristic is usually easier than synthesizing a two-terminal device with a prescribed driving-point characteristic. In this paper we propose an approach to synthesize a driving-point characteristic of a two-terminal device from the transfer characteristic of a two-port device, so that the resulting DP plot of the two-terminal device is exactly the same as the TC plot of the two-port device. We also illustrate the use of digital circuitry to synthesize arbitrary transfer characteristics. This technique will benefit the design and analysis of complex nonlinear electronic circuits and systems. A variety of characteristics synthesized using this approach are presented. Index Terms - Driving-point characteristics, nonlinear circuit synthesis, transfer characteristics.

In this brief, we study the possibility of coloring graphs by means of synchronized coupled oscillators. We consider an array of coupled oscillators as a graph by associating the oscillators to vertices and the coupling to edges. When the coupled array is synchronized, the phase of the oscillators can be considered as the color associated with the corresponding vertices. We prove that for connected 2-colorable graphs, we can construct a coupled array which generates the 2-coloring for that graph. For the general case, numerical simulation results with connected 3-colorable graphs suggest that the coupled array of oscillators can color graphs with a small number of colors in most cases. Some complexity issues of the system and comparisons to antivoter models of graph coloring will be discussed. We also conjecture that the system can be used to approximate the star chromatic number of the graph. Index Terms - Analog circuits, arrays, complexity theory, graph theory, oscillators, parallel processing, phase synchronization.

In classical sampled quantization, the signal is sampled at discrete times and at discrete values, resulting in uncertainty of the signal amplitude. However, the sampling times and the boundaries of the quantization intervals are still assumed to be known with infinite precision. The aim of this paper is to study quantization and sampling when these quantities are not known with infinite precision by considering quantized sampling in a general framework as sampling with uncertainty in time and amplitude. We define the concept of a valid quantized sample and consider a quantized sampling of a signal as a collection of valid quantized samples. We show that for continuous signals, a set of valid quantized samples generates a secondary set of valid quantized samples. We illustrate that oversampling can reduce reconstruction errors because oversampling can reduce the uncertainty in the secondary quantized samples. In particular, these secondary quantized samples have uncertainty approaching zero as oversampling increases, provided the sampling time and quantization thresholds are known with infinite precision. For a class of T-periodic bandlimited signals, this implies that the reconstruction error is a function of the oversampling ratio, the uncertainty in the sampling time, the stepsize of the quantizer, and the uncertainty in the quantization thresholds. Index Terms - Band-limited signals, jitter, oversampling, quantization, sampling methods, uncertainty.

Coupled oscillators are complicated high-dimensional dynamical systems. They can exhibit a wide variety of rich dynamics which could lead to novel applications in engineering. In this brief we describe a torus-doubling phenomenon observed from four mutually coupled Chua's circuits. The qualitative dynamical behavior of the coupled system is robust, yet the exact behavior is very sensitive to the initial conditions and the parameter values of the Chua's circuits. We present numerical simulation results from the system model which are in good qualitative agreement with the experimental measurements. Index Terms - Chua's circuits, coupled oscillators, torus-doubling bifurcation.

Cellular neural/nonlinear networks (CNN) are considered here for efficient implementation of the most computationally intensive steps of dynamic image coding. Several analogic CNN algorithms are presented for the generation of binary image masks and image decomposition. Measurement results for the first CNN universal chips executing an analogic algorithm for a reconstruction operator are also presented. Based on measured execution times, the viability of the CNN implementation of efficient but computationally expensive compression algorithms such as dynamic image coding is assessed. Index Terms - Computing architecture, image coding, image segmentation, morphology, multimedia, neural networks.

In this letter, a new chaos-based secure communication scheme is proposed in an attempt to thwart the attacks proposed recently. Instead of encoding the message signal in a chaotic system directly, we use two chaotic signals in our scheme. One of the chaotic signals is used to synchronize the chaotic encrypter and the chaotic decrypter. The other is used to encrypt the plain signal by using a multishift cipher scheme. Thus the transmitted signal is not used to encrypt the message and a more complicated method of encryption is used.

In this paper we present more analytical results about the complex behavior of a second order digital filter with overflow nonlinearity. We explore the parameter space to obtain a taxonomy of the different behaviors that occurs. In particular, we give a complete description of the chaotic behavior of the map F (which models the second order digital filter) in the parameter space (a, b). We prove that in the region [[bar R].sub.5] (the closure of [R.sub.5], where [R.sub.3] = {(a, b) : b < -q + 1. b < a + 1. b > -1}) F is not chaotic; in the region [absolute value of b] < 1 and (a, b) [not an element of] [[bar R].sub.5], F has a generalized hyperbolic attractor; and in the region [absolute value of b] > 1, if (a, b) are integers and b = -2(a - 1), then F is an exact map. In addition, we obtain some results concerning the fractal behavior of the map F. We find an estimate of the Hausdorff dimension of the generalized hyperbolic attractor. We obtain results regarding the symbolic dynamics of F. For example, we prove that the set of points with aperiodic admissible sequences in the case [absolute value of a] < 2 and b = -1 is uncountable.

In this letter, we prove several results regarding linear conjugacy between Lur'e systems. For example, we prove that except for a measure zero set, two Lur'e systems are linearly conjugate if they share an equilibrium point and the eigenvalues of the Jacobian matrices are matched at every point. A corollary of that result is that piecewise-linear vector fields with parallel boundary planes are determined, up to linear conjugacy, by the boundary planes, the equilibrium points and the eigenvalues in each region.

In this letter, we give supporting evidence for a conjecture regarding the amount of coupling needed to synchronize an array of linearly coupled dynamical systems. Roughly speaking, the conjecture says that the coupling needed to synchronize an array of coupled systems is inversely proportional to the nonzero eigenvalue of the coupling graph that is smallest in magnitude. The conjecture implies that the coupling needed to synchronize an array can be derived from the coupling topology and the coupling needed to synchronize two coupled dynamical systems.

In a companion paper, we have shown how Chua's oscillator is topologically conjugate to a class of 3-D systems. In this paper, we will use this result to approximate other chaotic systems in the literature which are not necessarily piecewise linear. To further illustrate the complexity of Chua's oscillator, we also include a gallery of the many attractors found in Chua's oscillator.

In this introductory tutorial paper, we demonstrate the generality of Chua's oscillator in generating chaos and bifurcation phenomena by electronic laboratory experiments which illustrate the standard routes to chaos, and by giving a result which shows that Chua's oscillator can generate the same qualitative behavior as any member of a 21-parameter family C of continuous, odd-symmetric, piecewise-linear vector field in R cubed. This result is of fundamental importance because it unifies many previously published papers on chaotic circuits and systems (e.g. examples from Brockett, Sparrow, Arneodo, Nishio, Ogorzalek, etc.) under one umbrella, thereby obviating the need to analyze these circuits and systems as separate and unrelated systems. Indeed, every bifurcation and chaotic phenomena exhibited by any member of the family C is also exhibited by this universal circuit. In a companion paper, we show how the generality of Chua's oscillator can be used to approximate other chaotic systems in the literature which are not necessarily piecewise-linear.

In this introductory tutorial paper, we show how 1-D maps can be useful in analyzing experimentally the chaotic dynamics and bifurcations of circuits and systems. We illustrate this by means of Chua's circuit.

Delay-type general neural networks and cellular neural networks (CNN's) proved to be important in different application areas including motion detection. In this paper, several stability results and additional properties of the delay-type neural network and CNN dynamics are proved. As a typical example, it is proved that if the feedback and delayed feedback matrices A and [A.sup.T] are non-negative and the sum (A + [A.sup.T]) is irreducible, then the neural network is stable.

We present some further stability results for cellular neural networks with nonlinear delay-type templates. In particular, we show that there exists a globally asymptotically stable equilibrium point in CNN's with dominant nonlinear delay-type templates.

The purpose of this letter is to discuss and provide a more rigorous stability proof for CNN with symmetric feedback templates.

A two-layer continuous-time cellular neural network for finding the Radon transform of a binary image is presented. The functionality of this cellular neural network follows from the functionality of the connected component detector cellular neural network., The Radon transform is a mathematical tool for obtaining a one-dimensional function of intensity values of a binary image in discrete-space. To this end, a double-layer cellular neural network (CNN) for continuous-time operation is introduced. The functionality of the CNN is shown to be dependent on the functionality of the connected component detector CNN.

It has been shown that synchronization between two nonlinear systems can be studied as a control-theory problem. We show that this relationship can be extended to synchronization in arbitrary coupled arrays of nonlinear systems. In particular, we use several well-known stability conditions to obtain synchronization criteria in arbitrarily coupled arrays: the passivity criterion, the circle criterion and a result on observer design of Lipschitz nonlinear systems. We also study how these synchronization criteria depend on the topology of the coupled networks. In particular, we show that synchronization is improved by using nonlocal connections or introducing random connections. Index Terms--Algebraic connectivity, arrays, circle criterion, nonlinear control, observer, passivity, random graphs, stability, synchronization of chaos.