Stochastic resonance in chua's circuit driven by alpha-stable noise

Thesis (Master)--Izmir Institute of Technology, Electronics and Communication Engineering, Izmir, 2012Includes bibliographical references (leaves: 75-80)Text in English; Abstract: Turkish and Englishx, 80 leavesThe main aim of this thesis is to investigate the stochastic resonance (SR) in Chua's circuit driven by alpha-stable noise which has better approximation to a real-world signal than Gaussian distribution. SR is a phenomenon in which the response of a nonlinear system to a sub-threshold (weak) input signal is enhanced with the addition of an optimal amount of noise. There have been an increasing amount of applications based on SR in various fields. Almost all studies related to SR in chaotic systems assume that the noise is Gaussian, which leads researchers to investigate the cases in which the noise is non-Gaussian hence has infinite variance. In this thesis, the spectral power amplification which is used to quantify the SR has been evaluated through fractional lower order Wigner Ville distribution of the response of a system and analyzed for various parameters of alpha-stable noise. The results provide a visible SR effect in Chua’s circuit driven by symmetric and skewed-symmetric alpha-stable noise distributions. Furthermore, a series of simulations reveal that the mean residence time that is the average time spent by the trajectory in an attractor can vary depending on different alpha-stable noise parameters

Memristor: A New Concept in Synchronization of Coupled Neuromorphic Circuits

The existence of the memristor, as a fourth fundamental circuit element, by researchers at Hewlett Packard (HP) labs in 2008, has attracted much interest since then. This occurs because the memristor opens up new functionalities in electronics and it has led to the interpretation of phenomena not only in electronic devices but also in biological systems. Furthermore, many research teams work on projects, which use memristors in neuromorphic devices to simulate learning, adaptive and spontaneous behavior while other teams on systems, which attempt to simulate the behavior of biological synapses. In this paper, the latest achievements and applications of this newly development circuit element are presented. Also, the basic features of neuromorphic circuits, in which the memristor can be used as an electrical synapse, are studied. In this direction, a flux-controlled memristor model is adopted for using as a coupling element between coupled electronic circuits, which simulate the behavior of neuron-cells. For this reason, the circuits which are chosen realize the systems of differential equations that simulate the well-known Hindmarsh-Rose and FitzHugh-Nagumo neuron models. Finally, the simulation results of the use of a memristor as an electric synapse present the effectiveness of the proposed method and many interesting dynamic phenomena concerning the behavior of coupled neuron-cells

A preliminary investigation into the effects of nonlinear response modification within coupled oscillators

This thesis provides an account of an investigation into possible dynamic interactions between two coupled nonlinear sub-systems, each possessing opposing nonlinear overhang characteristics in the frequency domain in terms of positive and negative cubic stiffnesses. This system is a two degree-of-freedom Duffing oscillator coupled in series in which certain nonlinear effects can be advantageously neutralised under specific conditions. This theoretical vehicle has been used as a preliminary methodology for understanding the interactive behaviour within typical industrial ultrasonic cutting components. Ultrasonic energy is generated within a piezoelectric exciter, which is inherently nonlinear, and which is coupled to a bar-horn or block-horn to one, or more, material cutting blades, for example. The horn/blade configurations are also nonlinear, and within the whole system there are response features which are strongly reminiscent of positive and negative cubic stiffness effects. The two degree-of-freedom model is analysed and it is shown that a practically useful mitigating effect on the overall nonlinear response of the system can be created under certain conditions when one of the cubic stiffnesses is varied. It has also bfeen shown experimentally that coupling of ultrasonic components with different nonlinear characteristics can strongly influence the performance of the system and that the general behaviour of the hypothetical theoretical model is indeed borne out in practice

Oscillation modes in multiresonant oscillator circuits

An in-depth analysis of the oscillation modes in free-running oscillators loaded with multiresonance networks is presented. The analysis illustrates the mechanisms for the generation and stabilization of the various periodic modes and establishes the conditions for existence of a single stable periodic mode in distinct regions of the parameter plane. The mechanisms for the generation and stabilization of quasi-periodic regimes, with two concurrent oscillations, are also analyzed, considering different situations in terms of two relevant poles. The stability analysis of quasi-periodic solutions, derived in terms of admittance functions, can be applied to circuits simulated with harmonic balance, under the assumption of high quality factor resonators. The impact of the transistor biasing on the stability properties of the quasi-periodic regimes has been analyzed, demonstrating that it can be used to isolate the quasi-periodic solution from the periodic ones. The analysis procedures have been applied to a practical oscillator based on two cross-coupled transistors at the two frequencies 900 MHz and 2.5 GHz. The case of two independent oscillations operating in a synchronized regime is also analyzed, as well as its impact on the phase-noise behavior.This work has been funded by the Spanish Government under contract TEC2014-60283-C3-1-R, the European Regional Development Fund (ERDF/FEDER) and the Parliament of Cantabria (12.JP02.64069)

Analysis and synthesis techniques of nonlinear dynamical systems with applications to diagnostic of controlled thermonuclear fusion reactors

Nonlinear dynamical systems are of wide interest to engineers, physicists and mathematicians, and this is due to the fact that most of physical systems in nature are inherently non-linear. The nonlinearity of these systems has consequences on their time-evolution, which in some cases can be completely unpredictable, apparently random, although fundamentally deterministic. Chaotic systems are striking examples of this. In most cases, there are no hard and fast rules to analyse these systems. Often, their solutions cannot be obtained in closed form, and it is necessary to resort to numerical integration techniques, which, in case of high sensitivity to initial conditions, lead to ill-conditioning problems and high computational costs. The dynamical system theory, the branch of mathematics used to describe the behaviour of these systems, focuses not on finding exact solutions to the equations describing the dynamical system, but rather on knowing if the system stabilises to a steady state in the long term, and what are the possible attractors, e.g. a quasi-periodic or chaotic attractors. Regarding the synthesis, from both a practical and a theoretical standpoint, it is very desirable to develop methods of synthesizing these systems. Although extensive theory has been developed for linear systems, no complete formulation for nonlinear systems synthesis is present today. The main topic of this thesis is the solution of engineering problems related to the analysis and synthesis of nonlinear and chaotic systems. In particular, a new algorithm which optimizes Lyapunov exponents estimation in piecewise linear systems has been applied to PWL and polynomial chaotic systems. In the field of complex systems synthesis, a systematic method to project systems of order 2n characterized by two positive Lyapunov exponents, has been proposed. This procedure couples nth-order chaotic systems with a suitable nonlinear coupling function. Furthermore, a method for the fault detection has been developed. In the field of time series analysis, a new denoising method, based on the wavelet transform of the noisy signal, has been described. The method implements a variable thresholding, whose optimal value is determined by analysing the cross-correlation between the denoised signal and the residuals and by applying different criteria depending on the particular decomposition level. Finally, a study of dynamical behaviour of Type I ELMs has been performed for a future modelization of the phenomenon. In this context, a statistical analysis of time intervals between successive Type I ELMs has been proposed.---------------------------------- Il tema principale di questa tesi è la soluzione di problemi ingegneristici legati all’analisi e alla sintesi di sistemi dinamici non lineari. I sistemi dinamici non lineari sono di largo interesse per ingegneri, fisici e matematici, e questo è dovuto al fatto che la maggior parte dei sistemi fisici in natura è intrinsecamente non lineare. La non linearità di questi sistemi ha conseguenze sulla loro evoluzione temporale, che in certi casi può rivelarsi del tutto imprevedibile, apparentemente casuale, seppure fondamentalmente deterministica. I sistemi caotici sono un esempio lampante di questo comportamento. Nella maggior parte dei casi non esistono delle regole standard per l’analisi di questi sistemi. Spesso, le soluzioni non possono essere ottenute in forma chiusa, ed è necessario ricorrere a tecniche di integrazione numerica, che, in caso di elevata sensibilità alle condizioni iniziali, portano a problemi di mal condizionamento e di elevato costo computazionale. La teoria dei sistemi dinamici, la branca della matematica usata per descrivere il comportamento di questi sistemi, non si concentra sulla ricerca di soluzioni esatte per le equazioni che descrivono il sistema dinamico, ma piuttosto sull’analisi del comportamento a lungo termine del sistema, per sapere se questo si stabilizzi in uno stato stabile e per sapere quali siano i possibili attrattori, ad esempio, attrattori quasi-periodici o caotici. Per quanto riguarda la sintesi, sia da un punto di vista pratico che teorico, è molto importante lo sviluppo di metodi in grado di sintetizzare questi sistemi. Sebbene per i sistemi lineari sia stata sviluppata una teoria ampia e esaustiva, al momento non esiste alcuna formulazione completa per la sintesi di sistemi non lineari. In questa tesi saranno affrontati problemi di caratterizzazione, analisi e sintesi, legati allo studio di sistemi non lineari e caotici. La caratterizzazione dinamica di un sistema non lineare permette di individuarne il comportamento qualitativo a lungo termine. Gli esponenti di Lyapunov sono degli strumenti che permettono di determinare il comportamento asintotico di un sistema dinamico. Essi danno informazioni circa il tasso di divergenza di traiettorie vicine, caratteristica chiave delle dinamiche caotiche. Le tecniche esistenti per il calcolo degli esponenti di Lyapunov sono computazionalmente costose, e questo fatto ha in qualche modo precluso l’uso estensivo di questi strumenti in problemi di grandi dimensioni. Inoltre, durante il calcolo degli esponenti sorgono dei problemi di tipo numerico, per ciò il calcolo deve essere affrontato con cautela. L’implementazione di algoritmi veloci e accurati per il calcolo degli esponenti di Lyapunov è un problema di interesse attuale. In molti casi pratici il vettore di stato del sistema non è disponibile, e una serie temporale rappresenta l’unica informazione a disposizione. L’analisi di serie storiche è un metodo di analisi dei dati provenienti da serie temporali che ha lo scopo di estrarre delle statistiche significative e altre caratteristiche dei dati, e di ottenere una comprensione della struttura e dei fattori fondamentali che hanno prodotto i dati osservati. Per esempio, un problema dei reattori a fusione termonucleare controllata è l’analisi di serie storiche della radiazione Dα, caratteristica del fenomeno chiamato Edge Localized Modes (ELMs). La comprensione e il 16 controllo degli ELMs sono problemi cruciali per il funzionamento di ITER, in cui il type-I ELMy H-mode è stato scelto come scenario di funzionamento standard. Determinare se la dinamica degli ELM sia caotica o casuale è cruciale per la corretta descrizione dell’ELM cycle. La caratterizzazione dinamica effettuata sulle serie temporali ricorrendo al cosiddetto spazio di embedding, può essere utilizzata per distinguere serie random da serie caotiche. Uno dei problemi più frequenti che si incontra nell’analisi di serie storiche sperimentali è la presenza di rumore, che in alcuni casi può raggiungere anche il 10% o il 20% del segnale. È quindi essenziale , prima di ogni analisi, sviluppare una tecnica appropriata e robusta per il denosing. Quando il modello del sistema è noto, l’analisi di serie storiche può essere applicata al rilevamento di guasti. Questo problema può essere formalizzato come un problema di identificazione dei parametri. In questi casi, la teorie dell’algebra differenziale fornisce utili informazioni circa la natura dei rapporti fra l’osservabile scalare, le variabili di stato e gli altri parametri del sistema. La sintesi di sistemi caotici è un problema fondamentale e interessante. Questi sistemi non implicano soltanto un metodo di realizzazione di modelli matematici esistenti ma anche di importanti sistemi fisici reali. La maggior parte dei metodi presentati in letteratura dimostra numericamente la presenza di dinamiche caotiche, per mezzo del calcolo degli esponenti di Lyapunov. In particolare, le dinamiche ipercaotiche sono identificate dalla presenza di due esponenti di Lyapunov positivi

Control of chaos in nonlinear circuits and systems

Nonlinear circuits and systems, such as electronic circuits (Chapter 5), power converters (Chapter 6), human brains (Chapter 7), phase lock loops (Chapter 8), sigma delta modulators (Chapter 9), etc, are found almost everywhere. Understanding nonlinear behaviours as well as control of these circuits and systems are important for real practical engineering applications. Control theories for linear circuits and systems are well developed and almost complete. However, different nonlinear circuits and systems could exhibit very different behaviours. Hence, it is difficult to unify a general control theory for general nonlinear circuits and systems. Up to now, control theories for nonlinear circuits and systems are still very limited. The objective of this book is to review the state of the art chaos control methods for some common nonlinear circuits and systems, such as those listed in the above, and stimulate further research and development in chaos control for nonlinear circuits and systems. This book consists of three parts. The first part of the book consists of reviews on general chaos control methods. In particular, a time-delayed approach written by H. Huang and G. Feng is reviewed in Chapter 1. A master slave synchronization problem for chaotic Lur’e systems is considered. A delay independent and delay dependent synchronization criteria are derived based on the H performance. The design of the time delayed feedback controller can be accomplished by means of the feasibility of linear matrix inequalities. In Chapter 2, a fuzzy model based approach written by H.K. Lam and F.H.F. Leung is reviewed. The synchronization of chaotic systems subject to parameter uncertainties is considered. A chaotic system is first represented by the fuzzy model. A switching controller is then employed to synchronize the systems. The stability conditions in terms of linear matrix inequalities are derived based on the Lyapunov stability theory. The tracking performance and parameter design of the controller are formulated as a generalized eigenvalue minimization problem which is solved numerically via some convex programming techniques. In Chapter 3, a sliding mode control approach written by Y. Feng and X. Yu is reviewed. Three kinds of sliding mode control methods, traditional sliding mode control, terminal sliding mode control and non-singular terminal sliding mode control, are employed for the control of a chaotic system to realize two different control objectives, namely to force the system states to converge to zero or to track desired trajectories. Observer based chaos synchronizations for chaotic systems with single nonlinearity and multi-nonlinearities are also presented. In Chapter 4, an optimal control approach written by C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao is reviewed. Systems with nonparametric regression with jump points are considered. The rough locations of all the possible jump points are identified using existing kernel methods. A smooth spline function is used to approximate each segment of the regression function. A time scaling transformation is derived so as to map the undecided jump points to fixed points. The approximation problem is formulated as an optimization problem and solved via existing optimization tools. The second part of the book consists of reviews on general chaos controls for continuous-time systems. In particular, chaos controls for Chua’s circuits written by L.A.B. Tôrres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes are discussed in Chapter 5. An inductorless Chua’s circuit realization is presented, as well as some practical issues, such as data analysis, mathematical modelling and dynamical characterization, are discussed. The tradeoff among the control objective, the control energy and the model complexity is derived. In Chapter 6, chaos controls for pulse width modulation current mode single phase H-bridge inverters written by B. Robert, M. Feki and H.H.C. Iu are discussed. A time delayed feedback controller is used in conjunction with the proportional controller in its simple form as well as in its extended form to stabilize the desired periodic orbit for larger values of the proportional controller gain. This method is very robust and easy to implement. In Chapter 7, chaos controls for epileptiform bursting in the brain written by M.W. Slutzky, P. Cvitanovic and D.J. Mogul are discussed. Chaos analysis and chaos control algorithms for manipulating the seizure like behaviour in a brain slice model are discussed. The techniques provide a nonlinear control pathway for terminating or potentially preventing epileptic seizures in the whole brain. The third part of the book consists of reviews on general chaos controls for discrete-time systems. In particular, chaos controls for phase lock loops written by A.M. Harb and B.A. Harb are discussed in Chapter 8. A nonlinear controller based on the theory of backstepping is designed so that the phase lock loops will not be out of lock. Also, the phase lock loops will not exhibit Hopf bifurcation and chaotic behaviours. In Chapter 9, chaos controls for sigma delta modulators written by B.W.K. Ling, C.Y.F. Ho and J.D. Reiss are discussed. A fuzzy impulsive control approach is employed for the control of the sigma delta modulators. The local stability criterion and the condition for the occurrence of limit cycle behaviours are derived. Based on the derived conditions, a fuzzy impulsive control law is formulated so that the occurrence of the limit cycle behaviours, the effect of the audio clicks and the distance between the state vectors and an invariant set are minimized supposing that the invariant set is nonempty. The state vectors can be bounded within any arbitrary nonempty region no matter what the input step size, the initial condition and the filter parameters are. The editors are much indebted to the editor of the World Scientific Series on Nonlinear Science, Prof. Leon Chua, and to Senior Editor Miss Lakshmi Narayan for their help and congenial processing of the edition