Solitons emerge in various non-linear systems as stable localized configurations, behaving in many ways like particles, from non-linear optics and condensed matter to nuclear physics, cosmology and supersymmetric theories. This book provides an introduction to integrable and non-integrable scalar field models with topological and non-topological soliton solutions. Focusing on both topological and non-topological solitons, it brings together debates around solitary waves and construction of soliton solutions in various models and provides a discussion of solitons using simple model examples. These include the Kortenweg-de-Vries system, sine-Gordon model, kinks and oscillons, and skyrmions and hopfions. The classical field theory of scalar field in various spatial dimensions is used throughout the book in presentation of related concepts, both at the technical and conceptual level. Providing a comprehensive introduction to the description and construction of solitons, this book is ideal for researchers and graduate students in mathematics and theoretical physics.

How is free will possible in the light of the physical and chemical underpinnings of brain activity and recent neurobiological experiments? How can the emergence of complexity in hierarchical systems such as the brain, based at the lower levels in physical interactions, lead to something like genuine free will? The nature of our understanding of free will in the light of present-day neuroscience is becoming increasingly important because of remarkable discoveries on the topic being made by neuroscientists at the present time, on the one hand, and its crucial importance for the way we view ourselves as human beings, on the other. A key tool in understanding how free will may arise in this context is the idea of downward causation in complex systems, happening coterminously with bottom up causation, to form an integral whole. Top-down causation is usually neglected, and is therefore emphasized in the other part of the book 's title. The concept is explored in depth, as are the ethical and legal implications of our understanding of free will.

This book arises out of a workshop held in California in April of 2007, which was chaired by Dr. Christof Koch. It was unusual in terms of the breadth of people involved: they included physicists, neuroscientists, psychiatrists, philosophers, and theologians. This enabled the meeting, and hence the resulting book, to attain a rather broader perspective on the issue than is often attained at academic symposia. The book includes contributions by Sarah-Jayne Blakemore, George F. R. Ellis, Christopher D. Frith, Mark Hallett, David Hodgson, Owen D. Jones, Alicia Juarrero, J. A. Scott Kelso, Christof Koch, Hans K ng, Hakwan C. Lau, Dean Mobbs, Nancey Murphy, William Newsome, Timothy O Connor, Sean A.. Spence, and Evan Thompson.

Our book presents a unique and original viewpoint on natural and engineered systems. The authors' goal is to propose and explain core principles that govern the formation and function of simple and complex systems. Examples are drawn from a broad range of topics from common materials and manufactured structures to the behavior of cells, organisms and socio-economic organizations. We provide a technical discussion of key engineering principles without the use of mathematics so that we may describe for a general audience how the systems of daily life form, operate, and evolve. We use analogy and illustrations to show how the components self-organize and scale to form complex adaptive systems. In this way we hope to understand how those systems come to be, achieve stability, and suddenly transition to new equilibrium states, including the sudden onset of economic recessions, ecosystem collapse, the evolution of species, development of cancer, and other wide-ranging topics. The existential role of component variability in these processes is emphasized.This book targets engineering instructors and undergraduate students curious to explore the grand challenges facing society today so they might build productive and long-lasting careers in science and technology. The six essays can be used to frame classroom discussions on systems from a broad range of disciplines. The essays are designed to appeal to those with a basic science and engineering background as we illustrate many fundamental engineering concepts in our descriptions of system behavior. We also hope our book appeals to curious members of the general public who are interested in understanding foundational ideas.

This book presents select, recent developments in nonlinear and complex systems reported at the 1st Online Conference on Nonlinear Dynamics and Complexity, held on November 23-25, 2020. It provides an exchange recent developments, discoveries, and progresses in Nonlinear Dynamics and Complexity. The collection presents fundamental and frontier theories and techniques for modern science and technology, stimulates more research interest for exploration of nonlinear science and complexity; and passes along new knowledge and insight to the next generation of engineers and technologists in a range of fields.

This book presents recent results and envisages new solutions of the stabilization problem for infinite-dimensional control systems. Its content is based on the extended versions of presentations at the Thematic Minisymposium "Stabilization of Distributed Parameter Systems: Design Methods and Applications" at ICIAM 2019, held in Valencia from 15 to 19 July 2019. This volume aims at bringing together contributions on stabilizing control design for different classes of dynamical systems described by partial differential equations, functional-differential equations, delay equations, and dynamical systems in abstract spaces. This includes new results in the theory of nonlinear semigroups, port-Hamiltonian systems, turnpike phenomenon, and further developments of Lyapunov's direct method. The scope of the book also covers applications of these methods to mathematical models in continuum mechanics and chemical engineering. It is addressed to readers interested in control theory, differential equations, and dynamical systems.

Our book presents a unique and original viewpoint on natural and engineered systems. The authors' goal is to propose and explain core principles that govern the formation and function of simple and complex systems. Examples are drawn from a broad range of topics from common materials and manufactured structures to the behavior of cells, organisms and socio-economic organizations. We provide a technical discussion of key engineering principles without the use of mathematics so that we may describe for a general audience how the systems of daily life form, operate, and evolve. We use analogy and illustrations to show how the components self-organize and scale to form complex adaptive systems. In this way we hope to understand how those systems come to be, achieve stability, and suddenly transition to new equilibrium states, including the sudden onset of economic recessions, ecosystem collapse, the evolution of species, development of cancer, and other wide-ranging topics. The existential role of component variability in these processes is emphasized.This book targets engineering instructors and undergraduate students curious to explore the grand challenges facing society today so they might build productive and long-lasting careers in science and technology. The six essays can be used to frame classroom discussions on systems from a broad range of disciplines. The essays are designed to appeal to those with a basic science and engineering background as we illustrate many fundamental engineering concepts in our descriptions of system behavior. We also hope our book appeals to curious members of the general public who are interested in understanding foundational ideas.

This volume examines current research in mechanics and its applications to various disciplines, with a particular focus on fluid-structure interaction (FSI). The topics have been chosen in commemoration of Dr. Bong Jae Chung and with respect to his wide range of research interests. This volume stands apart because of this diversity of interests, featuring an interdisciplinary and in-depth analysis of FSI that is difficult to find conveniently collected elsewhere in the literature. Contributors include mathematicians, physicists, mechanical and biomechanical engineers, and psychologists. This volume is structured into four thematic areas in order to increase its accessibility: theory, computations, experiments, and applications. Recent Advances in Mechanics and Fluid-Structure Interaction with Applications will appeal to established researchers as well as postdocs and graduate students interested in this active area of research.

"This book covers some of the main aspects of nonlinear analysis. It concentrates on stressing the fundamental ideas instead of elaborating on the intricacies of the more esoteric ones...it encompass es] many methods of dynamical systems in quite simple and original settings. I recommend this book to anyone interested in the main and essential concepts of nonlinear analysis as well as the relevant methodologies and applications." --MATHEMATICAL REVIEWS

The interplay between synchronization and spatio-temporal pattern formation is central for a broad variety of phenomena in nature, such as the coordinated contraction of heart tissue, associative memory and learning in neural networks, and pathological synchronization during Parkinson disease or epilepsy. In this thesis, three open puzzles of fundametal research in Nonlinear Dynamics are tackled: How does spatial confinement affect the dynamics of three-dimensional vortex rings? What role do permutation symmetries play in the spreading of excitation waves on networks? Does the spiral wave chimera state really exist? All investigations combine a theoretical approach and experimental verification, which exploit an oscillatory chemical reaction. A novel experimental setup is developed that allows for studying networks with N > 1000 neuromorphic relaxation oscillators. It facilitates the free choice of network topology, coupling function as well as its strength, range and time delay, which can even be chosen as time-dependent. These experimental capabilities open the door to a broad range of future experimental inquiries into pattern formation and synchronization on large networks, which were previously out of reach.

This twelfth volume in the Poincare Seminar Series presents a complete and interdisciplinary perspective on the concept of Chaos, both in classical mechanics in its deterministic version, and in quantum mechanics. This book expounds some of the most wide ranging questions in science, from uncovering the fingerprints of classical chaotic dynamics in quantum systems, to predicting the fate of our own planetary system. Its seven articles are also highly pedagogical, as befits their origin in lectures to a broad scientific audience. Highlights include a complete description by the mathematician E. Ghys of the paradigmatic Lorenz attractor, and of the famed Lorenz butterfly effect as it is understood today, illuminating the fundamental mathematical issues at play with deterministic chaos; a detailed account by the experimentalist S. Fauve of the masterpiece experiment, the von Karman Sodium or VKS experiment, which established in 2007 the spontaneous generation of a magnetic field in a strongly turbulent flow, including its reversal, a model of Earth's magnetic field; a simple toy model by the theorist U. Smilansky - the discrete Laplacian on finite d-regular expander graphs - which allows one to grasp the essential ingredients of quantum chaos, including its fundamental link to random matrix theory; a review by the mathematical physicists P. Bourgade and J.P. Keating, which illuminates the fascinating connection between the distribution of zeros of the Riemann -function and the statistics of eigenvalues of random unitary matrices, which could ultimately provide a spectral interpretation for the zeros of the -function, thus a proof of the celebrated Riemann Hypothesis itself; an article by a pioneer of experimental quantum chaos, H-J. Stoeckmann, who shows in detail how experiments on the propagation of microwaves in 2D or 3D chaotic cavities beautifully verify theoretical predictions; a thorough presentation by the mathematical physicist S. Nonnenmacher of the "anatomy" of the eigenmodes of quantized chaotic systems, namely of their macroscopic localization properties, as ruled by the Quantum Ergodic theorem, and of the deep mathematical challenge posed by their fluctuations at the microscopic scale; a review, both historical and scientific, by the astronomer J. Laskar on the stability, hence the fate, of the chaotic Solar planetary system we live in, a subject where he made groundbreaking contributions, including the probabilistic estimate of possible planetary collisions. This book should be of broad general interest to both physicists and mathematicians.

This book addresses a special topic in the field of nonlinear dynamical systems, develops a new research direction of surface chaos and surface bifurcation. It provides a clear watershed for original nonlinear chaos and bifurcation research. The novel content of this book makes nonlinear system research more systematical and personalized. This book introduces the chaos and bifurcation behavior of surface dynamics in the sense of Li Yorke, the basic properties, Lyapunov exponent and Feigenbaum constant of nonlinear behavior of surface, and obtained the wave behavior of chaotic process in surface motion, the control of surface chaos and bifurcation, and the wide application of surface chaos in engineering technology. Through this book, readers can obtain more abundant and novel contents about surface chaos and surface bifurcation than the existing mixed fitting bifurcation of plane curve and space curve, which can also expand the realm and vision of research.

This book provides readers with a detailed overview of second- and third-order nonlinearities in various nanostructures, as well as their potential applications. Interest in the field of nonlinear optics has grown exponentially in recent years and, as a result, there is increasing research on novel nonlinear phenomena and the development of nonlinear photonic devices. Thus, such a book serves as a comprehensive guide for researchers in the field and those seeking to become familiar with it. This text focuses on the nonlinear properties of nanostructured systems that arise as a result of optical wave mixing. The authors present a review of nonlinear optical processes on the nanoscale and provide theoretical descriptions for second and third-order optical nonlinearities in nanostructures such as carbon allotropes, metallic nanostructures, semiconductors, nanocrystals, and complex geometries. Here, the characterization and potential applications of these nanomaterials are also discussed. The factors that determine the nonlinear susceptibility in these systems are identified as well as the influence of physical mechanisms emerging from resonance and off-resonance excitations. In addition, the authors detail the effects driven by important phenomena such as quantum confinement, localized surface plasmon resonance, Fano resonances, bound states, and the Purcell effect on specific nanostructured systems. Readers are provided with a groundwork for future research as well as new perspectives in this growing field.

This book offers the first systematic account of canard cycles, an intriguing phenomenon in the study of ordinary differential equations. The canard cycles are treated in the general context of slow-fast families of two-dimensional vector fields. The central question of controlling the limit cycles is addressed in detail and strong results are presented with complete proofs. In particular, the book provides a detailed study of the structure of the transitions near the critical set of non-isolated singularities. This leads to precise results on the limit cycles and their bifurcations, including the so-called canard phenomenon and canard explosion. The book also provides a solid basis for the use of asymptotic techniques. It gives a clear understanding of notions like inner and outer solutions, describing their relation and precise structure. The first part of the book provides a thorough introduction to slow-fast systems, suitable for graduate students. The second and third parts will be of interest to both pure mathematicians working on theoretical questions such as Hilbert's 16th problem, as well as to a wide range of applied mathematicians looking for a detailed understanding of two-scale models found in electrical circuits, population dynamics, ecological models, cellular (FitzHugh-Nagumo) models, epidemiological models, chemical reactions, mechanical oscillators with friction, climate models, and many other models with tipping points.

This volume comprises the communications presented at the ETC 11, the EUROMECH European Turbulence conference held in 2007 in Porto.

The scientific committee has chosen the contributions out of the following topics: Acoustics of turbulent flows; Atmospheric turbulence; Control of turbulent flows; Geophysical and astrophysical turbulence; Instability and transition; Intermittency and scaling; Large eddy simulation and related techniques; MHD turbulence; Reacting and compressible turbulence; Transport and mixing; Turbulence in multiphase and non-Newtonian flows; Vortex dynamics and structure formation; Wall bounded flows.

The wave maps system is one of the most beautiful and challenging nonlinear hyperbolic systems, which has captured the attention of mathematicians for more than thirty years now. In the study of its various issues, such as the well-posedness theory, the formation of singularities, and the stability of the solitons, in order to obtain optimal results, one has to use intricate tools coming not only from analysis, but also from geometry and topology. Moreover, the wave maps system is nothing other than the Euler-Lagrange system for the nonlinear sigma model, which is one of the fundamental problems in classical field theory. One of the goals of our book is to give an up-to-date and almost self-contained overview of the main regularity results proved for wave maps. Another one is to introduce, to a wide mathematical audience, physically motivated generalizations of the wave maps system (e.g., the Skyrme model), which are extremely interesting and difficult in their own right.

This book presents a collection of problems for nonlinear dynamics, chaos theory and fractals. Besides the solved problems, supplementary problems are also added. Each chapter contains an introduction with suitable definitions and explanations to tackle the problems.The material is self-contained, and the topics range in difficulty from elementary to advanced. While students can learn important principles and strategies required for problem solving, lecturers will also find this text useful, either as a supplement or text, since concepts and techniques are developed in the problems.

This book provides an analysis of the construction, diagnosis (as chaotic) and evaluation of models in chaos theory. It contains a detailed look at the interaction of the different models used in chaos theory and analyses how these models influence the way chaos is defined. Furthermore, the book discusses the conditions for the occurrence of chaos and the detection of chaos in nature.

Most of the equations governing the problems related to science and engineering are nonlinear in nature. As a result, they are inherently difficult to solve. Analytical solutions are available only for some special cases. For other cases, one has no easy means but to solve the problem must depend on numerical solutions. Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes: Numerical Solutions presents the current theoretical developments of boundary layer theory, a branch of transport phenomena. Also, the book addresses the theoretical developments in the area and presents a number of physical problems that have been solved by analytical or numerical method. It is focused particularly on fluid flow problems governed by nonlinear differential equations. The book is intended for researchers in applied mathematics, physics, mechanics and engineering.

In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.

The second edition is substantially and formally revised and new material is added. Among this is bifurcation with a two-dimensional kernel with applications, the buckling of the Euler rod, the appearance of Taylor vortices, the singular limit process of the Cahn-Hilliard model, and an application of this method to more complicated nonconvex variational problems.

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Dynamics is what characterizes virtually all phenomenae we face in the real world, and processes that proceed in practically all kinds of inanimate and animate systems, notably social systems. For our purposes dynamics is viewed as time evolution of some characteristic features of the phenomenae or processes under consideration. It is obvious that in virtually all non-trivial problems dynamics can not be neglected, and should be taken into account in the analyses to, first, get insight into the problem consider, and second, to be able to obtain meaningful results.

A convenient tool to deal with dynamics and its related evolution over time is to use the concept of a dynamic system which, for the purposes of this volume can be characterized by the input (control), state and output spaces, and a state transition equation. Then, starting from an initial state, we can find a sequence of consecutive states (outputs) under consecutive inputs (controls). That is, we obtain a trajectory. The state transition equation may be given in various forms, exemplified by differential and difference equations, linear or nonlinear, deterministic or stochastic, or even fuzzy (imprecisely specified), fully or partially known, etc. These features can give rise to various problems the analysts may encounter like numerical difficulties, instability, strange forms of behavior (e.g. chaotic), etc.

This volume is concerned with some modern tools and techniques which can be useful for the modeling of dynamics. We focus our attention on two important areas which play a key role nowadays, namely automation and robotics, and biological systems. We also add some new applications which can greatly benefit from the availability of effective and efficient tools for modeling dynamics, exemplified by some applications in security systems.

This book deals with the theory of Poincaré--Birkhoff normal forms, studying symmetric systems in particular. Attention is focused on general Lie point symmetries, and not just on symmetries acting linearly. Some results on the simultaneous normalization of a vector field describing a dynamical system and vector fields describing its symmetry are presented and a perturbative approach is also used. Attention is given to the problem of convergence of the normalizing transformation in the presence of symmetry, with some other extensions of the theory. The results are discussed for the general case of dynamical systems and also for the specific Hamiltonian setting.

This monograph examines in detail models of neural systems described by delay-differential equations. Each element of the medium (neuron) is an oscillator that generates, in standalone mode, short impulses also known as spikes. The book discusses models of synaptic interaction between neurons, which lead to complex oscillatory modes in the system. In addition, it presents a solution to the problem of choosing the parameters of interaction in order to obtain attractors with predetermined structure. These attractors are represented as images encoded in the form of autowaves (wave memory). The target audience primarily comprises researchers and experts in the field, but it will also be beneficial for graduate students.

After a short introduction to the fundamentals, this book provides a detailed account of major advances in applying fractional calculus to dynamical systems. Fractional order dynamical systems currently continue to gain further importance in many areas of science and engineering. As with many other approaches to mathematical modeling, the first issue to be addressed is the need to couple a definition of the fractional differentiation or integration operator with the types of dynamical systems that are analyzed. As such, for the fundamentals the focus is on basic aspects of fractional calculus, in particular stability analysis, which is required to tackle synchronization in coupled fractional order systems, to understand the essence of estimators for related integer order systems, and to keep track of the interplay between synchronization and parameter observation. This serves as the common basis for the more advanced topics and applications presented in the subsequent chapters, which include an introduction to the 'Immersion and Invariance' (I&I) methodology, the masterslave synchronization scheme for partially known nonlinear fractional order systems, Fractional Algebraic Observability (FAO) and Fractional Generalized quasi-Synchronization (FGqS) to name but a few. This book is intended not only for applied mathematicians and theoretical physicists, but also for anyone in applied science dealing with complex nonlinear systems.

The aim of this Book is to give an overview, based on the results of nearly three decades of intensive research, of transient chaos. One belief that motivates us to write this book is that, transient chaos may not have been appreciated even within the nonlinear-science community, let alone other scientific disciplines.

This volume consists of 14 contributed chapters written by leading experts, offering in-depth discussions of the mathematical modeling and algorithmic aspects for tackling a range of space engineering applications. This book will be of interest to researchers and practitioners working in the field of space engineering. Since it offers an in-depth exposition of the mathematical modelling, algorithmic and numerical solution aspects of the topics covered, the book will also be useful to aerospace engineering graduates and post-graduate students who wish to expand their knowledge by studying real-world applications and challenges that they will encounter in their profession. Readers will obtain a broad overview of some of the most challenging space engineering operational scenarios of today and tomorrow: this will be useful for managers in the aerospace field, as well as in other industrial sectors. The contributed chapters are mainly focused on space engineering practice. Researchers and practitioners in mathematical systems modelling, operations research, optimization, and optimal control will also benefit from the case studies presented in this book. The model development and optimization approaches discussed can be extended towards other application areas that are not directly related to space engineering. Therefore, the book can be a useful reference to assist in the development of new modelling and optimization applications.