Nonlinear Dynamics of Electronic Systems

The following sections are included:

• Foreword

• Organization

• Contents

Usage of digital media has witnessed a tremendous growth during the last decades. However digital media are extremely vulnerable to copyright infringement, tampering and unauthorized distribution. Recently the protection of digital information has received significant attention within the digital media community and a number of techniques that try to address the problem by hiding appropriate information within digital media have been proposed. In this paper we present two applications of chaotic systems to watermarking of multimedia data, namely secure spatial scrambling of watermark patterns prior to their embedding in images and generation of 1-D or 2-D watermark signals.

We briefly dwell upon possible relationships between mixing properties and regenerative behaviour of ergodic dynamical system.

We review some of the methods of the multivalued operator theory and show two general classes of problems arising in the analysis and design of electronic circuits where these methods could be successfully applied. To the first class belong determination of driving point and transfer DC characteristics of active circuits. The second class of problems concerns so-called constrained systems where eg. several oscillator design problems can be located By examples we show how the proposed methods can be applied in practice and we show also that multivalued characteristics are typical in such problems.

In this paper, the fundamental matrix associated to a second-order filter with syllabic companding is formulated. To that aim, the concepts introduced in [1] are applied. It is shown that for almost any initial condition required for the solution of a Riccati differential equation which is needed in the formulation of the dynamic eigenvalues, a valid fundamental matrix is obtained. It is also shown that the dynamic eigenpairs do not define necessarily normal solutions when the system equations of the filter are periodic.

In this paper we present hyperchaotic n-scroll attractors which are generated from a generalized Matsumoto-Chua-Kobayashi circuit. Computer simulations for hyperchaotic 4-scroll and 6-scroll attractors are shown. A Lur'e representation of the generalized circuit is given. Based upon this Lur'e representation the new attractors are immediately applicable in synchronization schemes that have been developed for Lur'e systems such as robust H_∞ synchronization.

In the report the nonlinear system with pulse frequency modulation and feedback is represented. For the analysis of a system the digital mathematical model was used. The purpose of a research - study of a character of nonlinearity of the systems with a feedback with the help of diagrams of dot transformations and bifurcation maps. Consideration of conditions of existence of periodic and random oscillations of a system.

The main objective of this work is to provide a deep understanding of some nontrivial dynamical behaviour (periodic, quasiperiodic and homoclinic motion) related to the Takens-Bogdanov (double-zero eigenvalue of the linearization matrix) and Hopf-pitchfork bifurcation (a pair of imaginary eigenvalues and the third one zero) corresponding to some sections of a triple-zero eigenvalue bifurcation in the Chua's equation with a cubic nonlinearity.

In this paper we propose a novel approach for chaos-based communications where the (digital) information is encoded in one or more topological properties of a chaotic attractor. The potential advantage of this system is its intrinsic robustness against noise and distortions, and its amenability to both coherent and noncoherent detection.

We here address the problem of optimizing the performance of Chaos-Based DS-CDMA systems over selective channel by means of a rake receiver structure. Some formal arguments show that a non-conventional policy for the adaptation of its coefficients, which exploits the knowledge of the statistical properties of chaos-based spreading sequences, leads to significant improvement in the communication quality.

In this paper we introduce a non-coherent chaotic baseband spreading technique which is efficient with respect to the bit energy. This efficiency is due to a despreading or noise reduction filter which cancels the influence of the initial condition of the chaotic transmitter and reduces the inband noise power. As expected according to the BER - (Eb/No) performance the non-coherent chaotic spreading technique is lower-bounded by the code coherent technique but is in general better than the proposed DCSK scheme.

In many applications, such as the IEEE802.11-compliant Wireless Local Area Networks [1] (WLANs), the overall system performance is limited not by channel noise but by deep frequency-selective fading resulting from multipath effects. Due to their inherently wideband carriers, chaotic modulation schemes perform much better under these propagation conditions than conventional narrow-band ones. The robustness of the Frequency-Modulated Differential Chaos Shift Keying (FM-DCSK) system against multipath propagation has been evaluated in [2]. Due to its excellent multipath performance, the FM-DCSK communications system offers a potentially simple and cheap alternative to conventional spread spectrum techniques. To increase the channel capacity and decrease the probability of data collision, communications systems are often expected to offer a multiple access capability. This paper shows how the FM-DCSK technique can be used to develop a multiple access communications system and evaluates its robustness in this application.

We present an overview of the progress made using chaotic maps to model individual and aggregated self-similar traffic streams and in particular their impact on queue performance. Our findings show that the asymptotic behaviour of the queue is a function only of the tail of the ON active periods, and not of the Hurst parameter: two different self-similar traffic traces can have the same Hurst parameter but have a very different effect on the queue statistics.

The impact of the traffic assumption on a slotted non-persistent CSMA technique for WLAN systems is studied, by performing a comparison between a classical but not-realistic Poisson distribution and a self-similar one. The self-similar traffic generation is obtained by using a chaotic system, with structural parameters varied in order to have different self-similarity degrees and activity factors, and to permit a correct comparison with the Poisson traffic. The joint effect of the hidden terminals and capture phenomena has been taken into account, by considering a strictly dependence of the hidden terminals and capture probabilities from the number of active users in each slot, in order to make realistic the investigation. The system topology is centralized, so that a hub in a strategic position realizes packet relay functions. The analysis is performed by means of simulations.

Discrete-value and continuous-value maps are compared with respect to their cryptographical properties. In order to provide a more rigid evaluation of nonlinear chaotic encryption systems information-theoretic measures are derived that allow a comparison with classical discrete-value systems in relevant practical situations. Furthermore the consideration gives some insight about the general limitations of continuous-value operations in cryptographic applications. The application of these evaluation measures is illustrated by some examples.

In this paper a method to synchronize hyperchaotic cryptosystems via a scalar transmitted signal is proposed with the aim of enhancing the security of communication systems. The scalar signal is obtained as a linear combination of nonlinear functions and encrypted signals via two properly chosen squarewaves. This scheme enables to transmit alternatively the nonlinear functions of the driving system and the encrypted signals to the decrypter in a sequence of identical time frames. More in detail, a satisfactory synchronization between the encrypter and the decrypter can be obtained by considering these nonlinear functions. The suggested method is applied to bidirectionally coupled Chua's oscillators. Simulation results are carried out to highlight the performances of the proposed cryptosystem.

Globally Coupled Map (GCM) is one of the coupled systems of one-dimensional chaotic maps and is investigated in detail. However, there is no electrical circuit system that is suitable for comparison with GCM. In this study, we propose the system of N chaotic oscillators using Wien-bridge oscillators coupled by one resistor and investigate spatio-temporal phenomena for large N in order to compare with GCM. By carrying out numerical calculations, we observe various behaviors. In the investigated system, we can observe clustering, cluster bifurcation, chaotic itinerancy, and so on. From these observations, we denote the proposed system is very suitable for GCM model in electrical circuits.

Given a nonlinear electronic circuit, an associated linear time-varying small-signal circuit is formally derived by the tableau-method. It has the same topology as the original circuit while each original circuit element is replaced by an incremental one, evaluated along the signal-dependent nonlinear circuit solution. Since the variational circuit is linear in the first place, the designer is now invited to use the results of linear circuit theory. Furthermore, we present time-discrete companion models for linear time-varying elements. By them, the small-signal circuit equations can be efficiently solved numerically.

The frequency behaviour of time-varying small-signal models of nonlinear circuits is described by a modal expansion. Key concepts of time-invariant theory as normal mode, natural frequency and pole are generalized into the time-varying context. It is shown that with the exception of slowly-varying circuits, the accumulated meaning of a pole in time-invariant theory is lost for time-varying systems. Their impact turns out to be restricted exclusively to the high-frequency behaviour. Moreover, these high-frequency poles no longer coincide with the natural frequencies, while their location in the right-hand side of the complex frequency-plane neither predicts instability. Instead, their classic role in stability theory has to be replaced by a newly introduced complex Lya-punov exponent. This concept is closely related to the earlier introduced dynamic eigenvalue. The latter formally follows from a Riccati differential equation in its formed meaning of the generalized characteristic equation pertaining to linear time-varying systems. Both the complex Lyapunov exponent and the dynamic eigenvalue define to some extent a generalized pole concept.

The application of the Hopf Bifurcation Theorem (HBT) for maps is presented in this paper. The invariant cycle emerging from the bifurcation is approximated using an analogous version of the Graphical Hopf Theorem (GHT) for continuous-time systems. This technique is formulated in the so-called frequency domain and it involves the use of the Nyquist stability criterion for discrete-time systems. A neuronal netlet example is used for illustration.

In this paper, a new method for synthesizing nonlinear circuits is presented. One advantage of our approach is that we can directly synthesize nonlinear circuits from ordinary differential equations.

In this paper we present a method to find very long periodic orbits for one-dimensional maps. This new approach is a combination of the interval Newton method and the shooting technique. We also describe how to use this approach to find better approximation of position of computer generated pseudoperiodic orbit. Using this method we find very long periodic orbits for the logistic map and we calculate Lyapunov exponents of these orbits.

Recently, the authors have found wave-motion that phase states between adjacent oscillators change from in-phase to anti-phase or from anti-phase to in-phase in van der Pol oscillators coupled by inductors as a ladder. In this study, we make a simplified mathematical model for the ladder of van der Pol oscillators coupled by inductors in order to make clear the mechanism of the generation of the wave-motion. For the case of 9 oscillators, computer simulation of the model shows the wave-motion similar to that observed in the original circuit model, and we make clear the mechanism.

Mutual synchronizing areas of two Van der Pol type oscillators with "N"-shaped nonlinear conductances coupled by one inductor are studied. In order to apply averaging method in the proposed system simply, we use nonliner conductances which have a smooth "N"-shaped characteristic by a polynomial. As a results, we clarify that synchronizing area becames narrower when nonlinear conductances characteristics has higher order characteristics.

Chaotic regimes of a frequency controlled oscillator are investigated We show that, in a particular case, the mathematical model of such an oscillator coincides with the system describing Chua's circuit.

In this paper we discuss novel methods of classification and prediction of spatio-temporal dynamics in extended systems. We tested these methods on simulated data for the Kuramoto-Sivashinsky equation that describes unstable flame front propagation in uniform mixtures.

We propose novel generic RC realizations of Chua's circuit. Both realizations are based on the simplest possible models for second-order RC sinusoidal oscillators that are used to replace the active tank resonator in the classical Chua's circuit configuration. The sinusoidal oscillators are represented by circuit-independent black-box models. Hence, numerous circuit realizations can be derived.

An analytical approach for investigating free-oscillations in weakly non-linear systems is presented. It allows us to calculate the limit cycle in non-linear oscillators with shifting bias through an iterative procedure. The steady-state problem is formulated as a perturbation alternative problem, in the line of the functional analytic method of Cesari-Hale, which is iteratively solved by advantageously combining the classical perturbation method and the harmonic balance technique.

This paper proposes robust simulation of piecewise linear systems as a tool for the analysis of nonlinear electronic circuits. Rather than computing the evolution of a single trajectory, robust simulation computes the evolution from a set of initial conditions in the state space, for all forcing input signals within a given class. We describe here a tool to perform this analysis using mathematical programming. Among various applications, the tool allows to estimate the domain of attraction of equilibria, and to determine if some design specifications — expressed themselves in terms of reachability of subsets of the state-space — are met. A test of the tool on Chua's circuit is presented.

This paper considers Manifold Piecewise Linear system (ab. MPL) having partial constant return map. First, we consider a basic chaotic MPL which generates chaos. Its dynamics is governed by a 1-D return map. Second, applying the Instantaneous State Setting Method (ab. ISS) to the MPL, the chaotic return map is changed into a partial constant return map. Partial constant return map has a variety of Super Stable Periodic Orbits (ab. SSPO), and we have clarified their existence region.

In this paper the use of tools from optimal control theory, namely the Kalman filter, is introduced for chaotic Lur'e systems to replace the widely used error feedback synchronization. This allows to take into account observation noise on the strange attractor to be filtered We show that the filtering performance is superior to that of synchronization, in particular if the noise level is non-negligible.

We investigate the spectral properties of a simple sinusoidal signal frequency modulated by a PAM signal with random amplitude. We derive a general closed-form expression of the signal spectrum and show that whenever the modulation index is large enough, the shape of the spectrum is uniquely determined by the probability density function of the random symbols used in the PAM signal construction. Numerical simulation are reported, showing that the same results hold when the random variables are approximated by using the samples generated by a one-dimensional chaotic map. This contributes to give a first, partial, theoretical ground to the numerical and experimental results reported in [9][10][11].

This paper introduces into the theory of chaotic point processes (chaotic event streams). Processes of this type have been used in [1], [2], [3] and [4] for statistical signal modelling and analysis. Here a more systematic and comprehensive treatment of chaotic point processes is provided. The basic model is introduced and similarities to and differences from random point processes are shown. A basic statistical analysis is carried out. Approaches to the inverse problem (design) are discussed. Finally applications of chaotic point processes to the statistical analysis of chaos communication schemes and chaotic DC-DC-converters are presented.

This paper addresses the problem of synthesizing signals with prescribed statistical features using a causal system and in the particular case in which the instantaneous probability density and the correlation function (i.e. the power spectrum) are given.

The Frequency-Modulated Differential Chaos Shift Keying (FM-DCSK) modulation scheme offers a simple solution for wide-band communications applications such as indoor radio and Wireless Local Area Networks (WLAN). The main parameters of the telecommunications system have to be determined according to the special requirements of these applications. The most important system level parameters are the bandwidth of the channel filter, the bit duration, total radio frequency (RF) bandwidth of the transmitted signal and the Bit Error Rate (BER). This paper shows how these system level parameters have been determined for an FM-DCSK chaotic communications system which has been developed in the framework of a European Esprit project.

Over the past five years, many different chaotic modulation schemes have been developed. Unfortunately, exact analytical expressions for the evaluation of their noise performances are not yet available in the literature. This prevents the determination of theoretical performance bounds and the selection of optimum parameters for physical implementations of these systems. Introducing the basis function approach, this paper gives an exact description of chaotic modulation schemes and then exact closed form expressions for their noise performances. We show that the noise performance of coherent antipodal Chaos Shift Keying (CSK) and coherent Differential Chaos Shift Keying (DCSK) are equal to that of Binary Phase-Shift Keying (BPSK) and coherent Frequency-Shift Keying (FSK). If FM-DCSK is demodulated using the differentially coherent technique then its noise performance is just 3 dB worse than that of the conventional Differential Phase-Shift Keying (DPSK) modulation scheme demodulated with the differentially coherent technique.

The following sections are included:

• INTRODUCTION TO FM-DCSK SYSTEMS

• FULL- BILINEAR RECEIVERS

• UPPER BOUND ON COHERENT NOISE PERFORMANCE

• UPPER BOUND ON NON-COHERENT NOISE PERFORMANCE

• NUMERICAL RESULTS AND CONCLUSION

• REFERENCES

The paper describes the application of the Viterbi algorithm in the implmentation of a maximum likelihood receiver in a chaos communication scheme. The Viterbi decoder is constructed based on a symbolic state representation of the transmitter's chaos generator. The presented simulation results show that the classical Binary Phase Shift Keying (BPSK) modulation scheme can be slightly outperformed this way.

The object of the research is a two-dimensional first order recursive digital filter. Research of nonlinear effects originating in real digital filters is carried out. For analysing nonlinear effects both statistical and determined approaches are used. Within the statistical approach the statistical characteristics of quantization noise (expectation, variance) on the filter output are obtained. The upper estimations of average power of the quantization noise on the filter output are derived. Within the framework of the determined approach the filter with three levels' quantization is investigated. Overflow saturation nonlinearity is considered. Areas in space of system parameters corresponding to various regimes are obtained.

The paper presents some results from simulation of the sine-voltage driven series nonlinear RLC circuit in Figure 1 in the light of the study, in [1], of the existence or nonexistence of the true trajectories of chaotic dynamical systems that lie close to computer-generated trajectories. The nonexistence of true trajectories is caused by finite-time Lyapunov exponents fluctuating about zero. The circuit model used in the paper allows for a more in depth understanding of the structure of computer- generated chaotic trajectories.

In the paper we propose an alternative approach to information encryption using chaotic systems. In this approach binary data at the input generates binary data at the output, instead of analogue chaotic signal. The method joins two elements of secure signal generation: unique pseudorandom number sequence and chaotic system.

The paper presents the fully integrated design of the Wien-bridge chaos generator with the piecewise linear element built of MOS transistors. The circuit is implemented using SC technique. It is shown that this technique can be successfully used for the chaos oscillator implementation. The positive estimator of the Lyapunov exponent indicates the persistence of chaotic dynamics.

The problem of the analysis of nonlinear dynamical behaviour in the design of analogue circuits and systems is addressed in this paper. In contrast to the numerical simulations used in classical simulators, a method using an analytical behaviour description is considered here. This method is based on Volterra Series (VS). The use of VS in system analysis is introduced. An implementation of a system analysis tool based on VS is described. Complexity and efficiency of the implemented method is discussed an implementation details are presented. An example is given where the implemented analysis tool is used optimise a simple circuit. The results are verified with Spice. Finally, an outlook to open problems of the current implementation is given.

In addition to channel noise, the received signal in a radio communications system is corrupted by interferences. This paper evaluates by simulation the performance of a Frequency-Modulated Differential Chaos Shift Keying (FM-DCSK) modulation scheme when the radio channel suffers from strong interferences. First, the effect of another FM-DCSK radio signal transmitted in the same RF frequency band is considered. We show that the degradation in system performance depends on the bit durations of the two interfering FM-DCSK systems. The worst case occurs when the bit durations of the two systems coincide; a slight difference between the two bit durations reduces considerably the degradation in system performance. Then the FM-DCSK system is tested in an industrial environment where large spikes are present. These spikes cause large transients in the channel filter. Because an FM-DCSK signal can be demodulated by a nonlinear receiver, these transients can be eliminated by using a hard limiter in the receiver.

In Frequency-Modulated Differential Chaos Shift Keying (FM-DCSK) modulation, the digital information is carried by a wide-band chaotic signal. Since the FM-DCSK telecommunications system has been developed for Wireless Local Area Network (WLAN) application, the shape of the spectrum of the transmitted signal has to be optimized according to this application. The IEEE 802.11 standard for WLANs prescribe a spectrum which is free of any spikes, i.e., a transmitted signal without any strong periodic components. Another important requirement which has to be fulfilled in a WLAN is that the transmitted power in the adjacent channels has to be as low as possible in order to avoid co-channel interference. The paper analyzes the spectra of FM-DCSK signals and gives rules for the design of an optimum spectral shape to satisfy the WLAN specification.

Two types of dynamic math models of pulse phase-locked loop (PLL) with digital phase-frequency detector (PFD) are considered. One is a pulse model and another is a continuous-time approximation. They are used for acquisition analysis in the loop. A dynamic of non-linear PLL systems is simulated by PC. The solutions of continuous-time approximation equation are shown as figures. Dynamic features of PLL as non-linear mode system is discussed and PFD inertia is considered.

In this paper some results concerning synchronization of Chua circuits are presented. In particular, effects due to component tollerances in the two circuits, corresponding to parameter mismatch in the state equations, are investigated. Some measures of synchronizations are proposed and computed numerically for situations corresponding to different values of tollerance in electronic components, it is shown how component mismatch has some relevance in the onset of "bubbling" and may cause loss of synchronization.

The Lung sounds are currently classified as normal or adventitious, with the later being frequently associated to respiratory malfunctions. The adventitious sounds can be further divided into two classes: continuous and discontinuous. In this paper we propose the largest time dependent Lyapunov exponent (LTDLE) estimation as a new technique for pattern recognition applied to normal and adventitious lung sounds. It is shown that, both have associated positive LTDLEs, suggesting the presence of chaotic behaviours, with the largest Lyapunov exponent values associated to adventitious sounds and, among them, the continuous sounds exhibited the highest values. The results point out that the processes responsible for the lung sounds production have relevant non-linear components and that LTDLE may be a useful tool for lung sound analysis.

The purpose of the present paper is to extend the analysis of coupled oscillators to the case in which there is a small parameter mismatch. With this aim we apply and extend the approach developed by Johnson et al. [1] for unidirectional coupling. We show that synchronization in the neighborhood of the synchronization manifold for the identical subsystems can still occur, although the trajectories will exhibit characteristic fluctuations in the direction transverse to this manifold. The shift of the synchronization manifold and the amplitude of the transverse fluctuations are determined in terms of the coupling strength and the mismatch parameter. We illustrate these phenomena for two coupled Rössler systems and two coupled Sherman oscillators, describing the characteristic spiking behavior of pancreatic β-cells.

In this paper a novel approach for synchronizing chaotic systems with several positive Lyapunov exponents is described. The tool, which can be applied to high-order circuits as well as time-delay systems, enables synchronization to be achieved using a scalar signal.

Results of a qualitative analysis of an array of diffusively coupled identical continuous time dynamical systems are presented. The effect of partial chaotic synchronization are investigated via the linear invariant manifolds of the corresponding differential equations. Existence of various synchronization manifolds, a hierarchy and embedding of the manifolds of the coupled system are discovered. The general rigorous results are illustrated through examples of coupled Rössler systems.

Here we propose the mechanism for suppression of non-steady states, such as: propagation pulses, spiral waves, and spiral-waves chaos in excitable media. We shown that a short impulse change of parameters of the model can cause the de-stabilization and annihilation of excitation waves.

An improved model of a second-order Charge-Pump Phase-Locked Loop (CP-PLL) is proposed. An event-driven second-order CP-PLL macro-model is further developed from that described by Hedayat [1]. This model is made practical by taking account of VCO overload. Transient simulations are shown which illustrate the nature of phase-locking.

Evidence of chaotic behavior of electrical circuits can be divided into three categories [1]: a) laboratory experiments, b) computer simulations and c) mathematical proofs. As the author of [1] wrote, mathematical proofs for the chaotic nature of satisfactory model systems were rare in 1987, being the field of electrical circuits no exception. In his opinion, completely convincing evidence for chaotic behavior is only given by a), b) and c) together. In this communication, basic geometrical features will be described for a discontinuous system modelling a dc-dc buck converter under voltage control. The problem we are concerned with in this paper is to prove the existence of an attractor with an embedded horseshoe mechanism and discussing how it is created. The local analysis performed in [2] about the existence of strange attractors in the system is now complemented in a more geometrical way.

This paper shows how electrothermal dynamic interactions may cause the onset of stable oscillations in some common electronic circuits. These oscillations, predicted by a mixed state-variabie approach but not by conventional ones, may cause unexpected behaviour and, possibly, device destruction. Some examples have been observed both in simulations and in experiments.

A new type of Flash A/D architecture has been proposed in [3]. A comparison is made by chains of resistively coupled cells exhibiting wave propagation. In [4] an analysis was made on the meta-stable state of such a system. Here, we investigate the influence of the parameters of a chain on accuracy and speed. We analyse the effects of deviations and the design procedure for selecting the threshold levels. Insight is gained that helps selecting the parameters for the optimization of an A/D system for speed and accuracy. Finally conclusions are drawn with suggestions for further research.

Changing of dynamics of a system consisting of two Chua's circuits by introducing nonlinear coupling of nondiffusion type is investigated. In the case of unidirectional coupling, suppression of intrinsic activity of the element is observed. In the case of mutual couplings, an increase of the couplings leads to suppression of oscillations in the system and setting of a stationary state.

In this study, we propose the system that is two oscillators coupled by diodes for a simple designing technique. Relations between the ratio of eigenvalues of two oscillators coupled by diodes and the generation of chaotic attractors are investigated on three kinds of oscillators.

Bifurcation analysis of linear dynamical systems with relay feedback is considered. The emphasis is on periodic solutions and their symmetry. It is shown that, despite what has been conjectured in the literature, a symmetric and unforced relay feedback system can exhibit asymmetric periodic solutions. Moreover, the occurrence of periodic solutions characterized by one or more sections lying within the system discontinuity set is outlined. The mechanisms underlying their formation are carefully studied and shown to be due to an interesting, novel class of local bifurcations.

In addition to the stable fixed point which should be achieved under steady-state conditions, the sampling phase-locked loop (SPLL) implemented with a loop filter has another stable attractor in which an unwanted periodic solution, called false lock, develops in the loop. After the acquisition process, the SPLL either reaches the desired fixed point or gets into false lock, depending on the initial conditions. The chance of developing false lock must be avoided in practical SPLLs. The main goal of the paper is to give a model for the SPLL which can be used to explain how the system gets into false lock. The theoretical results have been verified by measurements.

Biological neocortical neural networks are essentially composed of identifiable microcircuits. We investigate a canonical microcircuit containing four neurons: two interacting pyramidal neurons N1, N2 that form the basic computational circuit, a (stellate) control neuron N3, and a pyramidal neuron N4 acting as a read-out device Using in-vitro experiments, we experimentally prove that under generic input conditions, the computationally most relevant neurons N1, N2 are on limit-cycle solutions. We propose a novel coding scheme, which is established through along Arnold tongues-organized neuron synchronization, and use it to estimate the computational properties of the microcircuit.

An approximated recursive convolution technique has been proposed to study lossy transmission lines loaded with nonlinear resistors. The introduced approach leads to a system of coupled discrete maps, each of them similar to the map obtained in case of nondispersive lines. By using the proposed technique a time-delayed Chua's circuit with a lossy line has been analysed.

This work deals with some laser models considered as slow-fast autonomous dynamical systems (S-FADS 1). Such systems consists in differential equations systems having a small parameter multiplying one or more velocity components. In order to analyze their solutions, some being chaotic, we have proposed a mathematical analytic method based on an iterative approach [1]. Under some conditions, this method allows us to give an analytic equation of the slow manifold. This equation is obtained by considering that the slow manifold is locally defined by the orthogonal plan to the tangent system's fast eigenvector.

It is also possible to compute the slow manifold equation by using the tangent system's slow eigenvectors. In some cases, this approach leads to an easier numerical plotting of the slow manifold.

This method allows us to give a geometrical characterization of the attractor and a global qualitative description of its dynamics.

[The contents of this paper will soon be presented in a more detailed publication in International Journal of Bifurcation and Chaos, and in collaboration with Prof. L. O. Chua and Prof. R. Lozi.].

The following sections are included:

• Author Index