Besides turbulence there is hardly any other scientific topic which has been considered as a prominent scientific challenge for such a long time. The special interest in turbulence is not only based on it being a difficult scientific problem but also on its meaning in the technical world and our daily life. This carefully edited book comprises recent basic research as well as research related to the applications of turbulence. Therefore, both leading engineers and physicists working in the field of turbulence were invited to the iTi Conference on Turbulence held in Bad Zwischenahn, Gemany 25th - 28th of September 2005. Discussed topics include, for example, scaling laws and intermittency, thermal convection, boundary layers at large Reynolds numbers, isotropic turbulence, stochastic processes, passive and active scalars, coherent structures, numerical simulations, and related subjects.

Besides turbulence, there is hardly any other scientific topic which has been considered a prominent scientific challenge for such a long time. The special interest in turbulence is not only based on it being a difficult scientific problem but also on its meaning in the technical world and our daily life. This carefully edited book comprises recent basic research as well as research related to the applications of turbulence. Therefore, both leading engineers and physicists working in the field of turbulence were invited to the iTi Conference on Turbulence held in Bad Zwischenahn, Gemany 21st - 24th of September 2003. Topics discussed include, for example, scaling laws and intermittency, thermal convection, boundary layers at large Reynolds numbers, isotropic turbulence, stochastic processes, passive and active scalars, coherent structures, numerical simulations, and related subjects.

Nature is inherently noisy and nonlinear. It is noisy in the sense that all macroscopic systems are subject to the fluctuations of their environments and also to internal fluctuations. It is nonlinear in the sense that the restoring force on a system displaced from equilibrium does not usually vary linearly with the size of the displacement. To calculate the properties of stochastic (noisy) nonlinear systems is in general extremely difficult, although considerable progress has been made in the past. The three volumes that make up Noise in Nonlinear Dynamical Systems comprise a collection of specially written authoritative reviews on all aspects of the subject, representative of all the major practitioners in the field. The first volume deals with the basic theory of stochastic nonlinear systems. It includes an historical overview of the origins of the field, chapters covering some developed theoretical techniques for the study of coloured noise, and the first English-language translation of the landmark 1933 paper by Pontriagin, Andronov and Vitt.

The theory of random dynamical systems originated from stochastic
differential equations. It is intended to provide a framework and
techniques to describe and analyze the evolution of dynamical
systems when the input and output data are known only approximately, according to some probability distribution. The development of this field, in both the theory and applications, has gone in many directions. In this manuscript we introduce measurable expanding random dynamical systems, develop the thermodynamical formalism and establish, in particular, the exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen s formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we arrive at a natural classification of the systems into two classes: quasi-deterministic systems, which share many
properties of deterministic ones; and essentially random systems, which are rather generic and never bi-Lipschitz equivalent to deterministic systems. We show that in the essentially random case the Hausdorff measure vanishes, which refutes a conjecture by Bogenschutz and Ochs.Lastly, we present applications of our results to various specific conformal random systems and positively answer a question posed by Bruck and Buger concerning the Hausdorff dimension of quadratic random Julia sets."

This volume contains the proceedings of the US-Australia workshop on Control and Chaos held in Honolulu, Hawaii from 29 June to 1 July, 1995. The workshop was jointly sponsored by the National Science Foundation (USA) and the Department of Industry, Science and Technology (Australia) under the US-Australia agreement. Control and Chaos-it brings back memories of the endless reruns of "Get Smart" where the good guys worked for Control and the bad guys were associated with Chaos. In keeping with current events, Control and Chaos are no longer adversaries but are now working together. In fact, bringing together workers in the two areas was the focus of the workshop. The objective of the workshop was to bring together experts in dynamical systems theory and control theory, and applications workers in both fields, to focus on the problem of controlling nonlinear and potentially chaotic systems using limited control effort. This involves finding and using orbits in nonlinear systems which can take a system from one region of state space to other regions where we wish to stabilize the system. Control is used to generate useful chaotic trajectories where they do not exist, and to identify and take advantage of useful ones where they do exist. A controller must be able to nudge a system into a proper chaotic orbit and know when to come off that orbit. Also, it must be able to identify regions of state space where feedback control will be effective.

This 4-volume compendium contains the verbatim hard copies of all color slides from the Chua Lecture Series presented at HP in Palo Alto, during the period from September 22 to November 24, 2015. Each lecture consists of 90 minutes, divided into a formal lecture, a discussion session, and an Encore of special trivia that the audience found mesmerizing.These lectures share some unique features of the classic Feynman Lectures on Physics, as much of the materials are presented in the unique style of the author, and the content is original as discovered or invented by the author himself. Unlike most technical books that suffer a notoriously short life span as their features could be superseded by superior models, this series of Chua lectures are intended to never be obsolete - many concepts and principles introduced are in fact new laws of nature, written in the language of sophomore-level mathematics, providing the foundation and the elan vital for initiating and nurturing future concepts and inventions.Volume I - covers everything that a researcher may want to know about memristors but is too afraid to ask.Volume II - shows that memristors can be either volatile or non-volatile, and effectively proving that synapses are non-volatile memristors, while action potentials are generated by locally-active memristors.Volume III - presents an overview of the fascinating phenomenon called chaos, while immersing the audience with the sights and sound of chaos from the Chua Circuit, invented in 1984 by Leon Chua, and has now become the standard textbook example of chaos exhibited by a real nonlinear electronic circuit, and not by computer simulations.Volume IV - surprises the audience with a new law of nature - dubbed the local activity principle, as discovered and proved mathematically in 1996 by Leon Chua. In particular, a Corollary of Chua's local activity theorem, dubbed the edge of chaos, is shown via insightful examples to be the originator of most complex phenomena, including intelligence, creativity, and deep learning. The edge of chaos is Mother Nature's tool for overcoming the tyranny of the second law of thermodynamics by providing an escape hatch for entropy to decrease over time. Indeed, the local activity principle which is profusely illustrated in the final volume, is widely recognized as a new law of thermodynamics, and is identified as the sine qua non of all complex phenomena, including life itself.Exclusive Access to the accompanying Video and Audio materials comes with the purchase of this book.

This 4-volume compendium contains the verbatim hard copies of all color slides from the Chua Lecture Series presented at HP in Palo Alto, during the period from September 22 to November 24, 2015. Each lecture consists of 90 minutes, divided into a formal lecture, a discussion session, and an Encore of special trivia that the audience found mesmerizing.These lectures share some unique features of the classic Feynman Lectures on Physics, as much of the materials are presented in the unique style of the author, and the content is original as discovered or invented by the author himself. Unlike most technical books that suffer a notoriously short life span as their features could be superseded by superior models, this series of Chua lectures are intended to never be obsolete - many concepts and principles introduced are in fact new laws of nature, written in the language of sophomore-level mathematics, providing the foundation and the elan vital for initiating and nurturing future concepts and inventions.Volume I - covers everything that a researcher may want to know about memristors but is too afraid to ask.Volume II - shows that memristors can be either volatile or non-volatile, and effectively proving that synapses are non-volatile memristors, while action potentials are generated by locally-active memristors.Volume III - presents an overview of the fascinating phenomenon called chaos, while immersing the audience with the sights and sound of chaos from the Chua Circuit, invented in 1984 by Leon Chua, and has now become the standard textbook example of chaos exhibited by a real nonlinear electronic circuit, and not by computer simulations.Volume IV - surprises the audience with a new law of nature - dubbed the local activity principle, as discovered and proved mathematically in 1996 by Leon Chua. In particular, a Corollary of Chua's local activity theorem, dubbed the edge of chaos, is shown via insightful examples to be the originator of most complex phenomena, including intelligence, creativity, and deep learning. The edge of chaos is Mother Nature's tool for overcoming the tyranny of the second law of thermodynamics by providing an escape hatch for entropy to decrease over time. Indeed, the local activity principle which is profusely illustrated in the final volume, is widely recognized as a new law of thermodynamics, and is identified as the sine qua non of all complex phenomena, including life itself.Exclusive Access to the accompanying Video and Audio materials comes with the purchase of this book.

The apparent contradiction of the results of the Fermi-Pasta-Ulam experiment conducted in 1953 and 1954 with the hypothesis that essentially any nonlinearity would lead to a system exhibiting ergodic behaviour has become known as the Fermi-Pasta-Ulam Problem. This volume reviews the current understanding of this paradox without trying to force coherence on differing perspectives on the same problem by various groups or approaches. The contributions comprise studies of one-dimensional chains, descriptions of numerical methods, heuristic theories, addressing the long standing and controversial problem of distinguishing chaos from noise in signal analysis, metastability, the relation of the FPU motions with the integrable equations, approaches using methods of perturbation theory and the proof of the applicability of KAM theory in FPU chains with energy very close to a minimum. For the convenience of the reader the original work of FPU is reprinted in an appendix. The order of the contributions reflects the aim of leading the interested but inexperienced reader through gradual understanding, starting from general analysis, and proceeding towards more specialized topics."

This book is drawn from across many active fields of mathematics and physics. It has connections to atmospheric dynamics, spherical codes, graph theory, constrained optimization problems, Markov Chains, and Monte Carlo methods. It addresses how to access interesting, original, and publishable research in statistical modeling of large-scale flows and several related fields. The authors explicitly reach around the major branches of mathematics and physics, showing how the use of a few straightforward approaches can create a cornucopia of intriguing questions and the tools to answer them.

Since the publication of our first book [80], there has been a real resiu-gence of interest in the study of almost automorphic functions and their applications ([16, 17, 28, 29, 30, 31, 32, 40, 41, 42, 46, 51, 58, 74, 75, 77, 78, 79]). New methods (method of invariant s- spaces, uniform spectrum), and new concepts (almost periodicity and almost automorphy in fuzzy settings) have been introduced in the literature. The range of applications include at present linear and nonlinear evolution equations, integro-differential and functional-differential equations, dynamical systems, etc...It has become imperative to take a bearing of the main steps of the the ory. That is the main purpose of this monograph. It is intended to inform the reader and pave the road to more research in the field. It is not a self contained book. In fact, [80] remains the basic reference and fimdamental source of information on these topics. Chapter 1 is an introductory one. However, it contains also some recent contributions to the theory of almost automorphic functions in abstract spaces. VIII Preface Chapter 2 is devoted to the existence of almost automorphic solutions to some Unear and nonUnear evolution equations. It con tains many new results. Chapter 3 introduces to almost periodicity in fuzzy settings with applications to differential equations in fuzzy settings. It is based on a work by B. Bede and S. G. Gal [40].

This volume mainly deals with the dynamics of finitely valued sequences, and more specifically, of sequences generated by substitutions and automata. Those sequences demonstrate fairly simple combinatorical and arithmetical properties and naturally appear in various domains. As the title suggests, the aim of the initial version of this book was the spectral study of the associated dynamical systems: the first chapters consisted in a detailed introduction to the mathematical notions involved, and the description of the spectral invariants followed in the closing chapters.

This approach, combined with new material added to the new edition, results in a nearly self-contained book on the subject. New tools - which have also proven helpful in other contexts - had to be developed for this study. Moreover, its findings can be concretely applied, the method providing an algorithm to exhibit the spectral measures and the spectral multiplicity, as is demonstrated in several examples. Beyond this advanced analysis, many readers will benefit from the introductory chapters on the spectral theory of dynamical systems; others will find complements on the spectral study of bounded sequences; finally, a very basic presentation of substitutions, together with some recent findings and questions, rounds out the book.

This book presents the mathematical foundations of systems theory in a self-contained, comprehensive, detailed and mathematically rigorous way. It is devoted to the analysis of dynamical systems and combines features of a detailed introductory textbook with that of a reference source. The book contains many examples and figures illustrating the text which help to bring out the intuitive ideas behind the mathematical constructions.

The 1960s were perhaps a decade of confusion, when scientists faced d- culties in dealing with imprecise information and complex dynamics. A new set theory and then an in?nite-valued logic of Lot? A. Zadeh were so c- fusing that they were called fuzzy set theory and fuzzy logic; a deterministic system found by E. N. Lorenz to have random behaviours was so unusual that it was lately named a chaotic system. Just like irrational and imaginary numbers, negative energy, anti-matter, etc., fuzzy logic and chaos were gr- ually and eventually accepted by many, if not all, scientists and engineers as fundamental concepts, theories, as well as technologies. In particular, fuzzy systems technology has achieved its maturity with widespread applications in many industrial, commercial, and technical ?elds, ranging from control, automation, and arti?cial intelligence to image/signal processing, patternrecognition, andelectroniccommerce.Chaos, ontheother hand, wasconsideredoneofthethreemonumentaldiscoveriesofthetwentieth century together with the theory of relativity and quantum mechanics. As a very special nonlinear dynamical phenomenon, chaos has reached its current outstanding status from being merely a scienti?c curiosity in the mid-1960s to an applicable technology in the late 1990s. Finding the intrinsic relation between fuzzy logic and chaos theory is certainlyofsigni?cantinterestandofpotentialimportance.Thepast20years have indeed witnessed some serious explorations of the interactions between fuzzylogicandchaostheory, leadingtosuchresearchtopicsasfuzzymodeling of chaotic systems using Takagi-Sugeno models, linguistic descriptions of chaotic systems, fuzzy control of chaos, and a combination of fuzzy control technology and chaos theory for various engineering pract

Quantum Chaos provides a comprehensive overview of our understanding of chaotic behaviour in a wide variety of quantum and semiclassical systems, and describes both experimental and theoretical investigations. A general introduction sets out the main features of chaos in quantum systems. Thereafter, in an authoritative collection of papers, prominent scientists put forward their particular interpretations of quantum chaos, with reference to a broad range of interesting physical systems. As yet, there is no universally accepted definition of quantum chaos. However, by dealing with such a wide range of topics from different branches of physics, this book provides an overview of this rapidly expanding field, and will be of great interest to graduate students and researchers in many areas of physics and chemistry.

The theme of the first Abel Symposium was operator algebras in a wide sense. In the last 40 years operator algebras have developed from a rather special discipline within functional analysis to become a central field in mathematics often described as "non-commutative geometry." It has branched out in several sub-disciplines and made contact with other subjects. The contributions to this volume give a state-of-the-art account of some of these sub-disciplines and the variety of topics reflect to some extent how the subject has developed. This is the first volume in a prestigious new book series linked to the Abel prize.

This book attempts to give a presentation of the advance of our knowledge of phase portraits of quadratic systems, paying special attention to the historical development of the subject. This the only book that organizes the portraits into classes, using the notions of finite and infinite multiplicity and finite and infinite index. Classifications of phase portraits for various classes are given using the well-known methods of phase plane analysis.

Perturbation theory and in particular normal form theory has shown strong growth in recent decades. This book is a drastic revision of the first edition of the averaging book. The updated chapters represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are survey appendices on invariant manifolds. One of the most striking features of the book is the collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with illuminating diagrams.

Thank heavens for Jens Wittenburg, of the University of Karlsruhe in Germany. Anyone who 's been laboring for years over equation after equation will want to give him a great big hug. It is common practice to develop equations for each system separately and to consider the labor necessary for deriving all of these as inevitable. Not so, says the author. Here, he takes it upon himself to describe in detail a formalism which substantially simplifies these tasks.

This book presents a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. An informal introduction expounds the ubiquity of such models via numerous. The results are presented in an informal style, and illustrated with many examples. The book is aimed at a wide audience of applied mathematicians, engineers and scientists at the beginning postgraduate level. Almost no mathematical background is assumed other than basic calculus and algebra.

Mixing processes occur in many technological and natural applications, with length and time scales ranging from the very small to the very large. The diversity of problems can give rise to a diversity of approaches. Are there concepts that are central to all of them? Are there tools that allow for prediction and quantification? The authors show how a variety of flows in very different settings possess the characteristic of streamline crossing. This notion can be placed on firm mathematical footing via Linked Twist Maps (LTMs), which is the central organizing principle of this book. The authors discuss the definition and construction of LTMs, provide examples of specific mixers that can be analyzed in the LTM framework and introduce a number of mathematical techniques which are then brought to bear on the problem of fluid mixing. In a final chapter, they present a number of open problems and new directions.

In this book, several world experts present (one part of) the mathematical heritage of Kolmogorov. Each chapter treats one of his research themes or a subject invented as a consequence of his discoveries. The authors present his contributions, his methods, the perspectives he opened to us, and the way in which this research has evolved up to now. Coverage also includes examples of recent applications and a presentation of the modern prospects.

Wave Turbulence refers to the statistical theory of weakly nonlinear dispersive waves. There is a wide and growing spectrum of physical applications, ranging from sea waves, to plasma waves, to superfluid turbulence, to nonlinear optics and Bose-Einstein condensates. Beyond the fundamentals the book thus also covers new developments such as the interaction of random waves with coherent structures (vortices, solitons, wave breaks), inverse cascades leading to condensation and the transitions between weak and strong turbulence, turbulence intermittency as well as finite system size effects, such as "frozen" turbulence, discrete wave resonances and avalanche-type energy cascades. This book is an outgrow of several lectures courses held by the author and, as a result, written and structured rather as a graduate text than a monograph, with many exercises and solutions offered along the way. The present compact description primarily addresses students and non-specialist researchers wishing to enter and work in this field.

This book provides an introduction to discrete dynamical systems - a framework of analysis that is commonly used in the ?elds of biology, demography, ecology, economics, engineering, ?nance, and physics. The book characterizes the fundamental factors that govern the quantitative and qualitative trajectories of a variety of deterministic, discrete dynamical systems, providing solution methods for systems that can be solved analytically and methods of qualitative analysis for those systems that do not permit or necessitate an explicit solution. The analysis focuses initially on the characterization of the factors that govern the evolution of state variables in the elementary context of one-dimensional, ?rst-order, linear, autonomous systems. The f- damental insights about the forces that a?ect the evolution of these - ementary systems are subsequently generalized, and the determinants of the trajectories of multi-dimensional, nonlinear, higher-order, non- 1 autonomous dynamical systems are established. Chapter 1 focuses on the analysis of the evolution of state variables in one-dimensional, ?rst-order, autonomous systems. It introduces a method of solution for these systems, and it characterizes the traj- tory of a state variable, in relation to a steady-state equilibrium of the system, examining the local and global (asymptotic) stability of this steady-state equilibrium. The ?rst part of the chapter characterizes the factors that determine the existence, uniqueness and stability of a steady-state equilibrium in the elementary context of one-dimensional, ?rst-order, linear autonomous systems.

H. Brezis: Proprietes regularisantes de certains semigroupes et applications.- F. Browder: Normal solvability and existence theorems for nonlinear mappings in Banach spaces.- F. Browder: Normal solvability for nonlinear mappings and the geometry of Banach spaces.- J. Eells, K.D. Elworthy: Wiener integration on certain manifolds.- W.H. Fleming: Nonlinear partial differential equations - Probabilistic and game theoretic methods.- C. Foias: Solutions statistiques des equations d'evolution non lineaires.- J.L. Lions: Quelques problemes de la theorie des equations non lineaires d'evolution.- A. Pazy: Semi-groups of nonlinear contractions in Hilbert space.- R. Temam: Equations aux derivees partielles stochastiques.- M.M. Vainberg: Le probleme de la minimisation des fonctionnelles non lineaires.

An extension of different lectures given by the authors, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems provides the reader with a comprehensive overview of these topics.

Starting with the simplest bifurcation problems arising for ordinary differential equations in one- and two-dimensions, this book describes several tools from the theory of infinite dimensional dynamical systems, allowing the reader to treat more complicated bifurcation problems, such as bifurcations arising in partial differential equations. Attention is restricted to the study of local bifurcations with a focus upon the center manifold reduction and the normal form theory; two methods that have been widely used during the last decades.

Through use of step-by-step examples and exercises, a number of possible applications are illustrated, and allow the less familiar reader to use this reduction method by checking some clear assumptions. Written by recognised experts in the field of center manifold and normal form theory this book provides a much-needed graduate level text on bifurcation theory, center manifolds and normal form theory. It will appeal to graduate students and researchers working in dynamical system theory.