This essay outlines the basic problems in reconstructing a state space attractor from a time series of data. One application, namely noise reduction, is outlined.

Now that the theory of nonlinear and chaotic systems is maturing, many people are beginning to consider applications of these phenomena. One application field which has become prominent is communications. This article is an overview of some of the recent advances in linking chaotic behavior to communications systems and signal processing.

We have shown that one may use chaotic signals to drive dynamical systems. When the driven system is stable to the driving signal and the driven system matches the system that produced the chaotic driving signal, the driven system will produce chaotic signals that are synchronized to the driving system. This may be seen in both autonomous and nonautonomous chaotic systems. This synchronized chaos may be useful in spread spectrum communications applications.

Work based upon the application of computer techniques for the study of safe basins of stability and Poincare sections for such devices has been extended to investigate the performance of high order models and also the relation to their forced performance. This paper combines a standard approach with methods based on ideas stimulated by chaotic solution. The effects of integrator leak, saturation, and nonlinear compensation techniques are also investigated with a view to chaotic behavior. Work previously due to Feely & Chua has also been extended to include higher order models.

Digital chaotic behavior in an optically processing element is reported. It is obtained as the result of processing two fixed train of bits. The process is performed with an Optically Programmable Logic Gate. Possible outputs for some specific conditions of the circuit are given. These outputs have some fractal characteristics, when input variations are considered. Digital chaotic behavior is obtained by using a feedback configuration. A random-like bit generator is presented.

It has been demonstrated that two identical chaotic systems can be made to synchronize by applying small, judiciously chosen, temporal parameter perturbations to one of them [Y. C. Lai and C. Grebogi, Phys. Rev. E 47, 2357(1993)]. This idea is applied to a nonlinear optical ring resonator modeled by the Ikeda-Hammel-Jones-Maloney map. The average time to achieve synchronization and the effect of noise are also discussed.

Sigma-Delta Modulation ((Sigma) (Delta) M) has attracted a great deal of interest as a method of analogue to digital conversion (ADC). This paper introduces a method of analysis for (Sigma) (Delta) M based on replacing the non-linear quantizer with a continuous element. Conventionally, analysis of sigma-delta modulation has been in the discrete domain, often treating the quantizer as a simple additive noise source, which has led to limited success in understanding the processes involved. However, the only truly discrete element in the circuit is the quantizer so if this could be represented accurately enough as a continuous element then a new form of analysis may be possible. A representation of the one-bit quantizer as a hyperbolic tangent function with a sufficiently steep gradient in the crossover region is proposed.

An innovative communication system has been developed. This system has the potential for improved secure communication for covert operations. By modulating data on the chaotic signal used to synchronize two nonlinear systems, we have created a Low Probability of Intercept (LPI) communications system. We derived the equations which govern the system. We made models of the system and performed numerical simulations to test these models. The theoretical and numerical studies of this system have been validated by experiment. A recent design improvement has led to a system that synchronizes at 0 db signal-to-noise ratio. This development holds the promise of a Low Probability of Detection (LPD) system.

The complexity inherent in the dynamics of chaotic oscillators can be utilized to produce digital communication signals. Symbolic (digital) information is encoded in large-scale features of the waveform by use of small perturbations to control the symbolic dynamics. A digital signal can thus be produced directly at the transmission stage, with no need for subsequent amplification. Because tiny perturbations control the dynamics, the circuitry for controlling the oscillator could often be entirely microelectronic. We first illustrate the main idea using the Lorenz system, which has a particularly simple symbolic dynamics description. We then describe other aspects of this mechanism for information transmission in the context of a previous paper.

Chaotic Frequency Modulation (CFM) provides the basis for a nonlinear communications system with (1) good noise suppression and (2) analogue signal encryption for secure communication links. A practical realization for a CFM transmitter employs an autonomous chaotic relaxation oscillator (ACRO) circuit for use as a chaotic voltage controlled oscillator (CVCO). The ACRO is simple to construct, consisting of only two capacitors, one inductor, a bistable nonlinear element, and a modulated current source. The CVCO period (P_k) is a nonlinear function of the current (m_k) and the two previous pulse periods. Demodulation requires the use of at least three successive waveform-periods. Experimental and theoretical studies of the CVCO circuit have shown that (1) the ACRO return maps of pulse periods are embedded in three dimensions, (2) chaotic outputs are difficult to decode without prior knowledge of the circuit parameters, and (3) demodulation may be accomplished with a digital signal processor.

We describe several existing and two new nonlinear dynamics (NLD)-based communication schemes, addressing the shortcomings and advantages of each. One new method is based on modulating the parameters of one or more chaotic generating systems with the signal(s) of interest. The output of the resulting nonstationary chaotic system is transmitted across the channel, and the signals are extracted via a continuous adaptation of system coefficients. The other novel method involves injecting the signal of interest as dynamic or feedback noise into the chaotic generating system. The noisy output is transmitted, and the dynamics of the generating system are estimated at the receiver. These dynamics are used to generate short term prediction errors, which are proportional to the original dynamic noise if the recovered system coefficients are reasonably accurate. Both methods have particular application to radio frequency (rf) communications, where spectral noise and frequency selective fading of signal power are real problems.

One of the most common methods by which a non-linear physical system can be determined to be stable, globally unstable, or in fact chaotic is by an investigation of its Lyapunov exponents. When the system contains a delay equation the correct co-ordinate space for the system is infinite dimensional. In this case special techniques must be used to make the problem solvable by numerical methods. We have investigated a practical example of such a retarded non-linear system: the continuous Sigma-Delta modulator (CSDM). The CSDM is an adaptation of the widely used digital Sigma-Delta modulator in which, now, the input and output are continuous functions of time, the quantizer has been replaced with a sharp hyperbolic tangent function and the feedback delay time can be varied continuously. This allows the system dynamics to be modeled by delay ordinary-differential equations. We have simulated such a device for various feedback configurations and delay times and have been able to show how a full calculation of the Lyapunov exponent spectrum allows a detailed analysis of stability conditions for the CSDM. Furthermore, we discuss the implications of this approach for retarded non-linear systems in general and a range of signal processing applications in particular.

Chaotic systems are known for their sensitivity to initial conditions. However, Pecora and Carroll have recently shown that a system, consisting of two Lorenz oscillators exhibiting chaos, could achieve synchronization, if a portion of the second system is driven by the corresponding portion of the first. It has been shown that the chaotic synchronization is related to asymptotic stability and that the method of the Lyapunov function can be used to prove synchronization, and to generate new systems exhibiting this phenomenon. In this paper, the main issue is that of the transfer of information between such synchronous chaotic systems. It has been shown that for a mutual transfer of information, a new type of synchronous organization is required. It leads to what we have termed as `the fraternal synchronization.' We have enumerated several interesting properties of fraternal synchronization, and followed it with a discussion of potential applications to parameter identification, communications and cryptography. Applications to biology and other fields are also briefly mentioned.

Channel identification is an important issue in communications in order to eliminate inter- symbol interference (ISI) and to overcome performance degradation due to fading channel or multipath propagation. A new method is proposed in this paper, which uses a chaotic sequence generated by a logistic map as a probe signal and estimate channel parameters according to dynamics of the chaotic sequence. The new method outperforms the technique where the least square (LS) method is used with a white Gaussian probe signal.

The method of lag-embedding, common in the analysis of signals in the context of nonlinear dynamics, requires the selection of an embedding dimension. This embedding dimension is analogous to the model order in a linear prediction model, but the order of a linear prediction model is of little use in characterizing chaotic signals or in indicating an appropriate embedding dimension for nonlinear analysis. Nonlinear prediction models, however, have been successfully used for this purpose. Here, we describe a technique for selecting an appropriate embedding dimension that is motivated by nonlinear prediction, but does not require the specification of the form of a prediction model.

This talk surveyed recent progress in forecasting nonlinear time series, including: the breakdown of linear systems theory, nonlinear generalizations by explicit embedding models and implicit neural network models, the role of geometrical invariants in embedding observations, using embedding to build forecasting models, the connection between high- dimensional entropy and dynamics in characterizing nonlinear systems, extension of embedding to more general linear transformations and to its experimental measurement by time-average expectation values, the difference between long-term modeling and short-term forecasting, and strategies for bringing time into connectionist architectures. The details of these techniques, as well as comparisons of their performance on a range of experimental data sets, is available in Time Series Prediction: Forecasting the Future and Understanding the Past, edited by Andreas Weigend and Neil Gershenfeld, Santa Fe Institute Studies in the Sciences of Complexity, Addison-Wesley, August 1993.

We describe a new approach to discriminating between `smoothness' and randomness in a time series. The method, which is applicable to both map and flow data, exploits an arbitrariness in the choice of vector field for computation of a statistic forming the basis of the test. Our approach mitigates uncertainties due to finite numerics in the statistics of previous work. We examine some of the examples chosen to illustrate a variety of effects that can occur in the implementation of this test.

Remarkable progress has been made in the detection of broad-band and possibly chaotic signals. Methods derived from the study of chaotic systems are also useful in the detection and analysis of certain man-made signals that are not inherently chaotic but are broad-band. These signals include phase-shift-keyed (PSK) communications signals that are used in many communications systems. Chaotic detection methods offer the prospect of detection of traditional broad-band signals, as well as the prospect of detection of many of the proposed chaotic communications schemes.

The natural structure of chaotic attracting sets generated by dynamic systems enables the encoding of some information in unstable periodic orbits embedded within the set. Unstable periodic orbits are a hidden invariant of chaotic dynamics, and information encoded in them is, accordingly, hidden. A method for stabilization of unstable periodic orbits of a chaotic attracting set by a weak parametric perturbation of the control parameter has been considered on the example of the Rossler attractor. By this method the hidden information can be extracted from a chaotic set. A hypothetical device of codal-lock-type based on this method is proposed. It is rigorously shown on the example of the logistic map that the action of periodic transformation in the space of parameters corresponding to its chaotic behavior leads to the stabilization of dynamics. As a result, certain functional coupling of two or more logistic maps with chaotic behavior generates stable periodic motion. On the basis of this a possible mechanism of transmitting secure information by chaotic signals is proposed.

With the advent of the studies of fractals, chaos, and non-linear dynamical processes by Mandelbrot, Falconer, Ruelle, and others, many of these results suggested to the researchers concerned with the mathematical modeling of non-linear physical behavior that there thus existed analytical bridges between the macroscopic and the microscopic world in the many scientific disciplines of particular concern. Of particular interest to the author is the use of irregular functions as forcing functions in the solution of such well known non-linear differential equations as Duffing's and Van der Pol's equations in the modeling of electrical or mechanical oscillations with temporarily changing frequencies. Irregular functions, as constructed by Weierstrass and Singh, are defined as those functions that are everywhere continuous, but nowhere differentiable. The use of Weierstrass's function as a forcing function for Duffing's non-linear equation is used to illustrate the behavior of a chaotic process generated by a frequency evolutionary process in time. A simple mechanical model of such a process is represented by the motion of a pendulum when the staff supporting the ball is shortened or lengthened during its execution of otherwise simple harmonic motion.

This research applies dynamical system methods (i.e., Chaos Theory) to the processing of time sequences of transitional and turbulent wall-pressures impinging on the face of station probes mounted along the wall of an axisymmetric body of revolution during a buoyant ascent from the bottom of a deep water test basin. It is demonstrated that the turbulent pressure fluctuations for this experiment can be described as a dynamical system of sufficiently low order (i.e., less than ten degrees of freedom). This opens up several possibilities for the control of turbulence. In underwater acoustics this translates to flow noise reduction in sonar applications and to drag reduction in ship dynamics. Other potential commercial applications include control of flow through pipelines, and aerodynamic design.

A cycle expansion method is applied to the calculation of a power spectrum of chaotic one- dimensional maps. It is shown that the broad-band part of the spectrum can be represented as a diffusion constant of some auxiliary process, and this constant is then represented in terms of periodic orbits. Accuracy of the method is also considered.

Ship navigation through ice-infested waters is a problem of deep concern to the oil exploration industry of the northern countries. Conventional marine radars do not perform satisfactorily in detecting small targets such as small pieces of iceberg. This paper reports a new method for detection in an ocean environment. The approach is based on the recent observation that sea clutter, radar echoes from the sea surface, can be modeled as a nonlinear deterministic dynamical system as an alternative to the conventional stochastic process. Based on this model, detection in sea clutter may be considered as classification of dynamical systems instead of statistical hypothesis testing. Two dynamical detection methods are proposed. The first one uses a dynamical invariant called attractor dimension to distinguish a target and a pure clutter process. The second approach tries to detect the existence of a target by observing the `difference' of the motion of the target and the clutter process. To show the validity of the idea of dynamical detection in sea clutter, real sea clutter and target data were used in this study.

An improved method for forecasting a time series is demonstrated. Forecasts are based on future behavior of nearest neighbors in an embedding space of time delay vectors. The metric of this space is generalized, rather than Euclidean. Reduction of forecast error and high probability of improved forecast are demonstrated for various chaotic time series. The approach also works for chaotic time series with added measurement noise. In addition, a lower bound on forecast error is calculated.

In this paper we utilize the gamma neural model to improve the signal to noise ratio (SNR) of broadband signals corrupted by white noise. The projection of a noisy signal onto the signal subspace cannot remove the noise in the subspace. A focus gamma network, when trained as a non-linear predictor of the projected trajectory, reduces this noise further. The property of adaptive memory depth of the gamma model is utilized to decide when to stop the training of the network. The preliminary results show that the SNR can be improved significantly, preserving the broadband signal spectrum.

A simple method describes how nonattracting chaotic sets can be reconstructed from time series by gluing pieces of many transiently chaotic signals together which come close to this invariant set. The method is illustrated with, and its validity is checked by, a map of well known dynamics, the Henon map. The nonattracting strange set is reconstructed in the presence of both a periodic and a chaotic attractor. Since the experimental investigation of transient chaos has received little attention, although the phenomenon is quite well understood theoretically, we hope that these findings might motivate further experimental studies.

Secure communication via chaotic synchronization of homogeneous driving systems of the Lorentz and Rossler type is demonstrated for AM, FM, SSB, and FSK modulations. Signal injection is additive. Actual AM, FM, SSB, and FSK signals were digitized using a 1 and 10 Mhz A/D. Analysis techniques of post chaotic modulation include the power spectrum, information dimension, Lyapunov exponents, mutual information, and metric entropy. Further inspection of the rates of convergence of the receiver subsystems gives insight into the selection of signal to driving system power ratio. Power ratios for adequate embedding for each modulation type in the chaotic signal are discussed. Differences in demodulation accuracy for the Lorentz and Rossler systems are discussed. It is seen that all modulation types can be sufficiently masked using a range of power ratio settings. However, the dynamical fidelity of the demodulated signal is compromised for some cases.

Chaotic time series analysis methods are applied to communications signals for characterization. Ergodic invariants of nonlinear physical processes are calculated for signals collected from AM, FM, FSK, and SSB radios. Previous work detailed information dimension and Lyapunov exponent calculations for these modulation types. The prevalence of positive Lyapunov exponents across modulation types indicated the presence of chaotic processes. Current results continue to indicate the appropriateness of this methodology for signal processing. Here, the results of the calculations of metric entropy, topological entropy, symbolic dynamics, recurrence maps, and circuit models are presented.

Mandelbrot, Falconer and others have demonstrated the existence of dimensionally invariant geometrical properties of non-linear dynamical processes known as fractals. Barnsley defines fractal geometry as an extension of classical geometry. Such an extension, however, is not mathematically trivial Of specific interest to those engaged in signal processing is the potential use of fractal geometry to facilitate the analysis of non-linear signal processes often referred to as non-linear time series. Fractal geometry has been used in the modeling of non- linear time series represented by radar signals in the presence of ground clutter or interference generated by spatially distributed reflections around the target or a radar system. It was recognized by Mandelbrot that the fractal geometries represented by man-made objects had different dimensions than the geometries of the familiar objects that abound in nature such as leaves, clouds, ferns, trees, etc. The invariant dimensional property of non-linear processes suggests that in the case of acoustic signals (active or passive) generated within a dispersive medium such as the ocean environment, there exists much rich structure that will aid in the detection and classification of various objects, man-made or natural, within the medium.

Chaos can be a useful feature imposed on the design of some electronic circuits and systems that need agility. A few unique circuits and systems are presented in which the use of chaos plays a definite design role. One unique application of chaos is in electromagnetic scattering, another is in advanced signal processing. More cutting edge utilitarian applications will come from emergent computation algorithms which `live on the edge of chaos' and use this agility as an evolutionary technique in transition to higher ordered states.

The wavelet transform is a bank of convolution filters indexed by scale; each scale of the transform of a signal is a filtered version of that signal. Here we explore the use of the Morlet wavelet to filter one coordinate of a dynamical system in order to visualize certain aspects of the geometry of the dynamics. This technique is a natural generalization of the differential phase plane representation.

Two methods of synchronization of chaotic oscillations in electronic circuits are considered. An example of application of synchronized chaos for secure communication is demonstrated by means of relaxation oscillators.

We have demonstrated experimentally a proportional feedback algorithm for the synchronization of chaotic time signals generated from a pair of independent diode resonator circuits. Synchronization was easily obtained and occurred for relative feedback levels between three and eight percent of the driving voltage. Once established, the synchronization persisted throughout the whole range of the resonator bifurcation diagram without varying the gain of the feedback.