Many dynamical systems in physics, chemistry and biology exhibit complex be­ haviour. The apparently random motion of a fluid is the best known example. How­ ever also vibrating structures, electronic oscillators, magnetic devices,lasers, chemical oscillators, and population kinetics can behave in a complicated manner. One can find irregular oscillations, which is now known as chaotic behaviour. The research field of nonlinear dynamical systems and especially the study of chaotic systems has been hailed as one of the important breaktroughs in science this century. The sim­ plest realization of a system with chaotic behaviour is an electronic oscillator. The purpose of this book is to provide a comprehensive introduction to the application of chaos theory to electronic systems. The book provides both the theoretical and experimental foundations of this research field. Each electronic circuit is described in detail together with its mathematical model. Controlling chaos of electronic oscilla­ tors is also included. End of proofs and examples are indicated by •. Inside examples the end of proofs are indicated with O. We wish to express our gratitude to Catharine Thompson for a critical reading of the manuscript. Any useful suggestions and comments are welcome. Email address of the first author: MVANWYK@TSAMAIL. TRSA. AC. ZA Email address of the first author: WHS@RAU3. RAU. AC. ZA Home page of the authors: http://zeus. rau. ac. za/steeb/steeb. html xi Chapter 1 Introduction 1.

This book explores the diverse types of Schrödinger equations that appear in nonlinear systems in general, with a specific focus on nonlinear transmission networks and Bose–Einstein Condensates. In the context of nonlinear transmission networks, it employs various methods to rigorously model the phenomena of modulated matter-wave propagation in the network, leading to nonlinear Schrödinger (NLS) equations. Modeling these phenomena is largely based on the reductive perturbation method, and the derived NLS equations are then used to methodically investigate the dynamics of matter-wave solitons in the network. In the context of Bose–Einstein condensates (BECs), the book analyzes the dynamical properties of NLS equations with the external potential of different types, which govern the dynamics of modulated matter-waves in BECs with either two-body interactions or both two- and three-body interatomic interactions.

This book provides a summary of the research conducted at UCLA, Stanford University, and UCSD over the last ?ve years in the area of nonlinear dyn- ics and chaos as applied to digital communications. At ?rst blush, the term “chaotic communications” seems like an oxymoron; how could something as precise and deterministic as digital communications be chaotic? But as this book will demonstrate, the application of chaos and nonlinear dynamicstocommunicationsprovidesmanypromisingnewdirectionsinareas of coding, nonlinear optical communications, and ultra-wideband commu- cations. The eleven chapters of the book summarize many of the promising new approaches that have been developed, and point the way to new research directions in this ?eld. Digital communications techniques have been continuously developed and re?ned for the past ?fty years to the point where today they form the heart of a multi-hundred billion dollar per year industry employing hundreds of thousands of people on a worldwide basis. There is a continuing need for transmission and reception of digital signals at higher and higher data rates. There are a variety of physical limits that place an upper limit on these data rates, and so the question naturally arises: are there alternative communi- tion techniques that can overcome some of these limitations? Most digital communications today is carried out using electronic devices that are essentially “linear,” and linear system theory has been used to c- tinually re?ne their performance. In many cases, inherently nonlinear devices are linearized in order to achieve a certain level of linear system performance.

Purpose and Emphasis. Mechanics not only is the oldest branch of physics but was and still is the basis for all of theoretical physics. Quantum mechanics can hardly be understood, perhaps cannot even be formulated, without a good kno- edge of general mechanics. Field theories such as electrodynamics borrow their formal framework and many of their building principles from mechanics. In short, throughout the many modern developments of physics where one frequently turns back to the principles of classical mechanics its model character is felt. For this reason it is not surprising that the presentation of mechanics reflects to some - tent the development of modern physics and that today this classical branch of theoretical physics is taught rather differently than at the time of Arnold S- merfeld, in the 1920s, or even in the 1950s, when more emphasis was put on the theoryandtheapplicationsofpartial-differentialequations. Today,symmetriesand invariance principles, the structure of the space–time continuum, and the geom- rical structure of mechanics play an important role. The beginner should realize that mechanics is not primarily the art of describing block-and-tackles, collisions of billiard balls, constrained motions of the cylinder in a washing machine, or - cycle riding.

For almost 15 years chaos and fractals have been riding a wave that has enveloped many areas of mathematics and the natural sciences in its power, creativity and expanse. Traveling far beyond the traditional bounds of mathematics and science to the distant shores of popular culture, this wave captures the attention and enthusiasm of a worldwide audience.

This volume of the Encyclopedia of Complexity and Systems Science, Second Edition, provides an authoritative introduction and overview of the latest research in cellular automata (CA) models of physical systems, emergent phenomena, computational universality, chaos, growth phenomena, phase transitions, self-organised criticality, reaction-diffusion systems, self-replications, parallel computation, and more.

This volume contains the proceedings of the NATO Advanced Research Workshop on `Quantum Chaos – Theory and Experiment', held at the Niels Bohr Institute, University of Copenhagen, from 28 May to 1 June 1991.

In this book we summarize the essential results of our efforts over the years to calculate energies, wave functions, and electromagnetic transitions of atoms as functions of the magnetic field strength from laboratory fields up to neutron star magnetic fields. Motivated by the observational evidence of huge magnetic 5 fields with strengths up to 10 T in the vicinity of white dwarf stars and of up 9 to 10 T in the vicinity of neutron stars the authors, together with coworkers and candidates for doctor and diploma degrees, have investigated this ,fasci­ nating quantum mechanical problem more or less continuously since 1978. The extensive tables and figures in the appendices represent the most complete data set to date in this field of research. For practical use all numbers are available by "anonymous ftp" over Internet. The first direct measurement of a neutron star magnetic field by Trum­ per and his group, who observed a cyclotron feature at about 50 ke V in the spectrum of the X-ray pulsar Hercules X-I corresponding to a field strength of 8 several 10 T, stimulated investigations of atoms within the framework of the adiabatic approximation, which is well justified for such field strengths. This method and its results are discussed in Chaps. 3, 5, and 6.

The proceedings of the 8th International Symposium on Chaos, Complexity, and Leadership present various concepts of diverse research methodologies to explore chaos and complexity across different scientific fields of leadership. Particular emphasis will be placed on investigating non-linearity to stimulate discussion on novel approaches and perspectives regarding chaos, complexity, and leadership.
The book delves into recent developments from diverse fields, employing chaos and complexity theory as analytical tools. Readers will find insights into important contemporary events, regardless of the various perspectives across social, political, economic, or scientific domains. It integrates numerous aspects and challenges regarding the impact of chaos and complexity on our world, utilizing nonlinear dynamics to suggest transformative policies. It aims to investigate and predict future paths through an interdisciplinary approach. Given the current societal and economic turbulence, leadership across all sectors necessitates reassessment. Consequently, this book primarily targets individuals interested in nonlinear science and leadership. However, its content extends its appeal to a broader audience due to its application of chaos and complexity to other disciplines.

Resonances are ubiquitous in dynamical systems with many degrees of freedom. They have the basic effect of introducing slow-fast behavior in an evolutionary system which, coupled with instabilities, can result in highly irregular behavior. This book gives a unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, a general finite dimensional theory of homoclinic jumping is developed and illustrated with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context. Previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds are described. The results are applied to a variety of different problems, which include applications from beam oscillations, surface wave dynamics, nonlinear optics, atmospheric science and fluid mechanics. The theory is further used to study resonances in Hamiltonian systems with applications to molecular dynamics and rigid body motion. The final chapter contains an infinite dimensional extension of the finite dimensional theory, with application to the perturbed nonlinear Schrödinger equation and coupled NLS equations.

This volume provides a comprehensive overview of chaos theory.

For almost 15 years chaos and fractals have been riding a wave that has enveloped many areas of mathematics and the natural sciences in its power, creativity and expanse. Traveling far beyond the traditional bounds of mathematics and science to the distant shores of popular culture, this wave captures the attention and enthusiasm of a worldwide audience.