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This book is devoted to the frequency domain approach, for both regular and degenerate Hopf bifurcation analyses. Besides showing that the time and frequency domain approaches are in fact equivalent, the fact that many significant results and computational formulas obtained in the studies of regular and degenerate Hopf bifurcations from the time domain approach can be translated and reformulated into the corresponding frequency domain setting, and be reconfirmed and rediscovered by using the fre...

This important book provides information necessary for those dealing with stochastic calculus and pricing in the models of financial markets operating under uncertainty; introduces the reader to the main concepts, notions and results of stochastic financial mathematics; and develops applications of these results to various kinds of calculations required in financial engineering. It also answers the requests of teachers of financial mathematics and engineering by making a bias towards probabilist...

Written for the non-specialist scientist wishing to bring chaos theory into focus as something conceptually and operationally useful in scientific applications, this text bridges the gap between non-mathematical popular treatments and the distinctly mathematical publications that non-mathematicians find so difficult to penetrate. The book provides derivations or explanations of many key concepts, such as Kolmogorov-Sinai entropy, dimensions, Fourier analysis and Lyapunov exponents. Only basic al...

Interacting chaotic oscillators are of interest in many areas of physics, biology, and engineering. In the biological sciences, for instance, one of the challenging problems is to understand how a group of cells or functional units, each displaying complicated nonlinear dynamic phenomena, can interact with one another to produce a coherent response on a higher organizational level.This book is a guide to the fascinating new concept of chaotic synchronization. The topics covered range from transv...

This volume is an outgrowth of the Third International Symposium on Hamiltonian Systems and Celestial Mechanics. The main topics are Arnold diffusion, central configurations, singularities in few-body problems, billiards, area-preserving maps, and geometrical mechanics. All papers in the volume went through the refereeing process typical of a mathematical research journal.

This festschrift is dedicated to Professor Howell Tong on the occasion of his 65th birthday. With a Foreword written by Professor Peter Whittle, FRS, it celebrates Tong's path-breaking and tireless contributions to nonlinear time series analysis, chaos and statistics, by reprinting 10 selected papers by him and his collaborators, which are interleaved with 17 original reviews, written by 19 international experts.Through these papers and reviews, readers will have an opportunity to share many of...

The motion of mechanical systems undergoing rotation about a fixed axis has been the subject of extensive studies over a few centuries. These systems are generally subject to gyroscopic forces which are associated with coriolis accelerations or mass transport and render complex dynamics.The unifying theme among topics presented in this book is the gyroscopic nature of the system equations of motion. The book represents comprehensive and detailed reviews of the state of art in four diverse applic...

This volume provides a self-contained survey of the mechanisms presiding information processing and communication. The main thesis is that chaos and complexity are the basic ingredients allowing systems composed of interesting subunits to generate and process information and communicate in a meaningful way. Emphasis is placed on communication in the form of games and on the related issue of decision making under conditions of uncertainty. Biological, cognitive, physical, engineering and societal...

This book is part of a two volume set which presents the analysis of nonlinear phenomena as a long-standing challenge for research in basic and applied science as well as engineering. It discusses nonlinear differential and differential equations, bifurcation theory for periodic orbits and global connections. The integrability and reversibility of planar vector fields and theoretical analysis of classic physical models are sketched. This first volume concentrates on the mathematical theory and...

In this book we describe the evolution of Classical Mechanics from Newton's laws via Lagrange's and Hamilton's theories with strong emphasis on integrability versus chaotic behavior.In the second edition of the book we have added historical remarks and references to historical sources important in the evolution of classical mechanics.

Chaos exists in systems all around us. Even the simplest system of cause and effect can be subject to chaos, denying us accurate predictions of its behaviour, and sometimes giving rise to astonishing structures of large-scale order. Our growing understanding of Chaos Theory is having fascinating applications in the real world - from technology to global warming, politics, human behaviour, and even gambling on the stock market. Leonard Smith shows that we all have an intuitive understanding of c...

The new science of chaos was discovered in the analysis of weather. According to the author, economics is equally unpredictable. In this book, the way chaos may be used for economicc analysis is explored. The author applies the new insights of chaotic dynamics to economics. Though the conception of chaos arose in abstract mathematics, it has already proved very fruitful in a number of applied sciences. Since it sets out to demonstrate that in a situation of total cause and effect, behaviour can...

This book deals with the qualitative theory of dynamical systems and is devoted to the study of flows and cascades in the vicinity of a smooth invariant manifold. Its main purpose is to present, as completely as possible, the basic results concerning the existence of stable and unstable local manifolds and the recent advancements in the theory of finitely smooth normal forms of vector fields and diffeomorphisms in the vicinity of a rest point and a periodic trajectory. A summary of the results o...

Dynamics of billiard balls and their role in physics have received wide attention since the monumental lecture by Lord Kelvin at the turn of the 19th century. Billiards can nowadays be created as quantum dots in the microscopic world enabling one to envisage the so-called quantum chaos, i.e. quantum manifestation of chaos of billiard balls. In fact, owing to recent progress in advanced technology, nanoscale quantum dots, such as chaotic stadium and antidot lattices analogous to the Sinai Billiar...

This book introduces the reader to the two main directions of one-dimensional dynamics. The first has its roots in the Sharkovskii theorem, which describes the possible sets of periods of all cycles (periodic orbits) of a continuous map of an interval into itself. The whole theory, which was developed based on this theorem, deals mainly with combinatorial objects, permutations, graphs, etc.; it is called combinatorial dynamics. The second direction has its main objective in measuring the complex...

This book covers the proceedings from the 2016 International Symposium on Chaos, Complexity and Leadership, and reflects current research results of chaos and complexity studies and their applications in various fields. Included are research papers in the fields of applied nonlinear methods, modeling of data and simulations, as well as theoretical achievements of chaos and complex systems. Also discussed are leadership and management applications of chaos and complexity theory.

Many complex systems - from immensely complicated ecosystems to minute assemblages of molecules - surprise us with their simple behaviour. Consider, for instance, the snowflake, in which a great number of water molecules arrange themselves in patterns with six-way symmetry. How is it that molecules moving seemingly at random become organized according to the simple, six-fold rule? How do the comings, goings, meetings and eatings of individual animals add up to the simple dynamics of ecosystem po...

Translates new mathematical ideas in nonlinear dynamics and chaos into a language that engineers and scientists can understand, and gives specific examples and applications of chaotic dynamics in the physical world. Also describes how to perform both computer and physical experiments in chaotic dynamics. Topics cover Poincare maps, fractal dimensions and Lyapunov exponents, illustrating their use in specific physical examples. Includes an extensive guide to the literature, especially that relati...

The thread of self-organization which is now recognized as permeating many dynamical transformations in diverse systems around us seems set to unleash a revolution as influential as that of Darwin in the last century. Darwin removed the 'originator' of a species; self-organization now seeks to remove the 'organizer' from an organism. Methods of nonlinear dynamics have played a crucial role in opening up this field and if these methods have a progenitor it is Henri Poi~care (1854 - 1912) whose fi...

Although individual orbits of chaotic dynamical systems are by definition unpredictable, the average behavior of typical trajectories can often be given a precise statistical description. Indeed, there often exist ergodic invariant measures with special additional features. For a given invariant measure, and a class of observables, the correlation functions tell whether (and how fast) the system “mixes”, i.e. “forgets” its initial conditions.This book, addressed to mathematicians and mathematica...

The 2021 IPCC report made one thing crystal clear - global climate change is here to stay. Time is up. We need to act or climate change will lead to inconceivable suffering by billions of people. Buying Time for Climate Action is the combined narrative of world class experts, all committed to help humanity survive its largely self-induced destructive course. Changing that course requires urgent action. Determining which actions will lead to helpful change requires insights into the stumbling blo...