Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.Comment: submitted to EP

Robust output synchronization for complex nonlinear systems.

Zhao, Jin.Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.Includes bibliographical references (leaves 79-83).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Synchronization of Master-slave Systems --- p.1Chapter 1.2 --- Output Regulation --- p.2Chapter 1.3 --- Typical Nonlinear Systems --- p.4Chapter 1.4 --- Organization --- p.4Chapter 2 --- Synchronization of Chua's Circuit and Van der Pol Oscillator via Inter- nal Model Approach --- p.6Chapter 2.1 --- Introduction --- p.6Chapter 2.2 --- Problem Formulation --- p.8Chapter 2.3 --- Preliminaries --- p.10Chapter 2.4 --- Solvability of the Problem --- p.13Chapter 2.4.1 --- The solution of the regulator equations --- p.14Chapter 2.4.2 --- Steady-state generator --- p.15Chapter 2.4.3 --- Internal model --- p.19Chapter 2.4.4 --- Stabilization --- p.20Chapter 2.4.5 --- Simulation --- p.22Chapter 2.5 --- Conclusions --- p.27Chapter 3 --- Robust Output Regulation of Output Feedback Systems with Nonlinear Exosystems --- p.28Chapter 3.1 --- Introduction --- p.28Chapter 3.2 --- Assumptions and Preliminaries --- p.29Chapter 3.3 --- Solvability of the Synchronization Problem --- p.33Chapter 3.4 --- Comparing Two Approaches for Output Regulation --- p.42Chapter 3.4.1 --- Differences between the two approaches for the output regulation problem --- p.42Chapter 3.4.2 --- Solvability of the regulator equations --- p.43Chapter 3.4.3 --- Solvability of stabilization --- p.47Chapter 3.5 --- Conclusions --- p.49Chapter 4 --- Applications of Robust Regional Synchronization via Output Regulation Techniques --- p.50Chapter 4.1 --- Problem Formulation --- p.50Chapter 4.2 --- Duffing Oscillator Synchronizes with Chua's Circuit --- p.51Chapter 4.2.1 --- Transfer the synchronization problem into the stabilization problem --- p.53Chapter 4.2.2 --- Boundedness of Chua's circuit --- p.57Chapter 4.2.3 --- Stabilization --- p.59Chapter 4.2.4 --- Simulation Results --- p.64Chapter 4.3 --- The Chaotic SMIB Power System Synchronizes with Van der Pol Oscillator --- p.64Chapter 4.3.1 --- Transfer the synchronization problem into the stabilization problem --- p.68Chapter 4.3.2 --- Stabilization --- p.71Chapter 4.3.3 --- Simulation Results --- p.74Chapter 4.4 --- Conclusions --- p.76Chapter 5 --- Conclusions --- p.77Bibliography --- p.7

Experimental Analysis of Emergent Dynamics in Complex Networks of Nonlinear Oscillators

The aim of this thesis is to explore and investigate the emergent dynamics of complex networks through a novel and insightful experimental setup realized as a configurable network of chaotic Chua's circuits. In particular part of our work has been devoted to the implementation and characterization of a "2.0 hardware version" of it, where the interconnection network has improved greatly in its main features. In this way the setup has been fully automatized in providing control on network structure and coupling strength. A large set of experiments has been carried out in networks with proportional coupling and arbitrary topology, showing, emergent dynamics encompassing synchronization, patterns and traveling waves, clusters formation. Also, the case of dynamic coupling has been experimentally addressed. The experimental observations have been compared with theoretical results by carrying out a local stability analysis of networks with static and dynamic links. Here we use the Master Stability approach (MSF) and its extensions to the case where the links are of dynamic nature (Proportional Derivative-MSF). Last part of the work has been devoted to the experimental study of cluster synchronization, stimulated by novel theoretical advances based on group theory and network symmetries. A novel network structure referred as "Multiplexed Network" has been experimentally examined, resulting in a great enhancement in synchronization, for which no theoretical models are yet available

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

Identification and prediction of nonlinear dynamics

Chaos theory and associated analyses are being applied to a growing number of disciplines. Studies of biological and ecological systems have shown the widest application of chaotic analyses thus far. When studying these systems, it is often only possible to measure a subset of the system\u27s many variables. To effectively perform a number of the analyses required to study a chaotic system, it is necessary to identify a complete strange attractor for the system. Consequently, it is necessary to reconstruct the system\u27s strange attractor from the available data. Many different methods exist for reconstructing strange attractors, but the effectiveness of each of these methods has not been studied and compared. This investigation examines the effectiveness of various reconstruction methods used to preserve the fractal structure of the attractor and the exponential divergence of nearby trajectories in an effort to determine the optimal method for reconstructing strange attractors. With an optimal method to reconstruct strange attractors for chaotic physical systems, engineers and scientists can more successfully characterize a nonlinear system and apply methods to predict its future behavior

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

Second order mem-circuits

This paper presents a comprehensive taxonomy of so-called second order memory devices, which include charge-controlled memcapacitors and flux-controlled meminductors, among other novel circuit elements. These devices, which are classified according to their differential and state orders, are necessary to get a complete extension of the family of classical nonlinear circuit elements (resistors, capacitors, inductors) for all possible controlling variables. Using a fully nonlinear formalism, we obtain nondegeneracy conditions for a broad class of second order mem-circuits. This class of circuits is expected to yield a rich dynamic behavior; in this regard we explore certain bifurcation phenomena exhibited by a family of circuits including a charge-controlled memcapacitor and a flux-controlled meminductor, providing some directions for future research

Second International Workshop on Harmonic Oscillators

The Second International Workshop on Harmonic Oscillators was held at the Hotel Hacienda Cocoyoc from March 23 to 25, 1994. The Workshop gathered 67 participants; there were 10 invited lecturers, 30 plenary oral presentations, 15 posters, and plenty of discussion divided into the five sessions of this volume. The Organizing Committee was asked by the chairman of several Mexican funding agencies what exactly was meant by harmonic oscillators, and for what purpose the new research could be useful. Harmonic oscillators - as we explained - is a code name for a family of mathematical models based on the theory of Lie algebras and groups, with applications in a growing range of physical theories and technologies: molecular, atomic, nuclear and particle physics; quantum optics and communication theory

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