A modular system for constructing dynamical systems

This thesis discusses a method based on the dual principle of Rössler, and developed by Deng, for systematically constructing robust dynamical systems from lower dimensional subsystems. Systems built using this method may be modified easily, and are suitable for mathematical modelling. Extensions are made to this scheme, which allow one to describe a wider range of dynamical behaviour. These extensions allow the creation of systems that reproduce qualitative features of the Lorenz Attractor (including bifurcation properties) and of Chua's circuit, but which are easily extensible

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

Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

Musical variations from a chaotic mapping

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1995.Includes bibliographical references (leaves 161-162).by Diana S. Dabby.Ph.D

Cone-like Invariant Manifolds for Nonsmooth Systems

This thesis deals with rigorous mathematical techniques for higher-dimensional nonsmooth systems and their applications. The dynamical behaviour of these systems is a nonlocal problem due to the lack of smoothness. Motivated by various examples of nonsmooth systems in applications, we propose to explore the concept of invariant surfaces in the phase space which is separated by a discontinuity hypersurface. For such systems the corresponding Poincaré map can be determined; it turns out that under suitable conditions an invariant cone occurs which is characterized by a fixed point of the Poincaré map. The invariant cone seems to serve in a similar way as a generalisation of the classical center manifold for smooth differential systems. Hence, the stability of the whole system can be reduced to investigate the stability on the two-dimensional surface of the cone. Motivated to study the generation of invariant cones out of smooth systems, a numerical procedure to establish invariant cones and their stability is presented. It has been found that the flat degenerate cone in a smooth system develops under nonsmooth perturbations into a cone-like configuration. Also a simple example is used to explain a paradoxical situation concerning stability. Theoretical results concerning the existence of invariant cones and possible mechanisms responsible for the observed behavior for general three dimensional nonsmooth systems are discussed. These investigations reveal that the system possesses a rich dynamic behavior and new phenomena such as, for instance, the existence of multiple invariant cones for such system. Our approach is developed to include the case when sliding motion takes place on the manifold. Sliding dynamical equations are formulated by using Filippov's method. Existence of invariant cones containing a segment of sliding orbits are given as well as stability on these cones. Different sliding bifurcation scenarios are treated by theoretical analysis and simulation. As an application we have investigated the dynamics of an automotive brake system model under the excitation of dry friction force which has served as a motivating example to develop our concepts. This model belongs to the class of nonsmooth systems of Filippov type which is investigated from direct crossing and a sliding motion point of view. Existence of invariant cones and different types of bifurcation phenomena such as sliding periodic doubling and multiple periodic orbits are observed. Finally, extensions to nonlinear perturbations of nonsmooth linear systems have been obtained by using the nonsmooth linear system as basic system. If the basic system possesses an attractive invariant cone without sliding motion, we have shown that locally the Poincaré map contains the necessary information with regard to attractivity of the invariant cone. The existence of a generalized center manifold reduction of nonlinear system has been proven by using Hadamard graph transformation approach. A class of nonlinear systems having a cone-like invariant "manifold" is presented to illustrate the center manifold reduction and associated bifurcation. The scientific contributions of parts of this thesis are presented in [32,39,66]

Análisis, construcción, simulación y sincronización de circuitos electrónicos prototipos de Caos

El proyecto tiene como objetivo el estudio de siete Sistemas Dinámicos, yendo de los que son paradigma de Caos a los más complejos, y sus posibles aplicaciones en comunicaciones privadas, bioingeniería y comunicaciones ópticas. El conjunto de sistemas seleccionados incluye algunos ejemplos paradigmáticos de Dinámicas Caóticas, así como nuevas propuestas, tanto de do sistemas básicos como de un sistema que tiene soluciones más complejas, nunca antes estudiados. Se logrará, de esta manera, realizar un completo recorrido desde los osciladores no-lineales más simples (como el de Van Der Pol), hasta los sistemas de mayor complejidad (como son las dinámicas hipercaóticas). El estudio consiste, en primer lugar, en identificar los métodos de análisis específicos del Caos, que permiten poner de manifiesto su carácter y propiedades (a lo que se dedicará el capítulo 1). Tras ello (Capítulo 2 y 3), se desarrollan, estudian y analizan los sistemas mediante simulaciones numéricas de la dinámica de los citados sistemas utilizando el software matemático MATLAB. En una segunda parte (que abarca la primera mitad del Capítulo 4), se implementan los circuitos electrónicos de los citados sistemas, y se simula su comportamiento mediante un software profesional. En una tercera parte (coincidente con la segunda mitad del Capítulo 4 y el Capítulo 5 completo), se construyen físicamente los sistemas fundamentales y sus extensiones, con el objetivo de caracterizar su comportamiento. Además, se desarrolla una aplicación software con entorno gráfico para el análisis sistemático de las dinámicas objeto de estudio. Finalmente, y con el objetivo de aplicar los Sistemas Dinámicos caóticos tanto a Comunicaciones Seguras como a Bioingeniería, este proyecto presenta un estudio de los citados sistemas para su uso en Comunicaciones Seguras, en el capítulo 6. Por otro lado, el oscilador de Van Der Pol no sólo es un sistema paradigma de Caos por la riqueza de su dinámica caótica, sino también por su interés en la simulación del corazón humano tanto en régimen regular, como en régimen caótico. Este análisis se desarrolla en el Capítulo 3

Analysis and synthesis of self-synchronizing chaotic systems

Includes bibliographical references (p. 225-228).Supported by the U.S. Air Force Office of Scientific Research. AFOSR-91-0034-C Supported by the U.S. Navy Office of Naval Research. N00014-91-C-0125 N00014-93-1-0686 Supported by Lockheed Sanders, Inc.Kevin M. Cuomo