Nonlinear science Books

Book SynopsisThis book presents a clear and concise introduction to the field of nonlinear dynamics and chaos, suitable for graduate students in mathematics, physics, chemistry, engineering, and in natural sciences in general. This second edition includes additional material and in particular a new chapter on dissipative nonlinear systems. The book provides a thorough and modern introduction to the concepts of dynamical systems' theory combining in a comprehensive way classical and quantum mechanical description. It is based on lectures on classical and quantum chaos held by the author at Heidelberg and Parma University. The book contains exercises and worked examples, which make it ideal for an introductory course for students as well as for researchers starting to work in the field.Table of ContentsIntroduction.- Fundamental terminology.- Complexity.- Classical versus quantum dynamics.- Problems.- References.- Dynamical systems.- Evolution law.- One-dimensional maps.- Problems.- References.- Nonlinear Hamiltonian systems.- Integrable examples.- Hamiltonian formalism.- Important techniques in the Hamiltonian formalism.- Integrable systems.- Non-integrable systems.- Perturbation of low-dimensional systems.- Canonical perturbation theory.- Transition to chaos in Hamiltonian systems.- Criteria for local and global chaos.- Appendix.- Problems.- References.- Dissipative systems - Introduction - Fixed points - Fixed point scenarios in two dimensional systems - Damped Oscillators - Harmonic oscillator - Nonlinear oscillators - Nonlinear damping - Poincaré-Bendixson Theorem - Damped forced oscillators - Driven one-Dimensional harmonic oscillator - Duffing oscillator - Lorenz model for turbulence - Fractals - Simple examples - Box-counting dimension - Examples from nature - Bifurcation scenarios - Examples of pitchfork bifurcations - Tangent bifurcations - Transcritical bifurcations - Higher-order bifurcations - Hopft bifurcations - Intermittency - Coupled Oscillators - Synchronisation - Kuramoto model - Increasing complexity - Problems - References. -Aspects of quantum chaos.- Introductory remarks on quantum mechanics.- Semiclassical quantization of integrable systems.- Semiclassical description of non-integrable systems.- Wave functions in phase space.- Anderson and dynamical localization.- Universal level statistics.- Concluding remarks.- Appendix.- Problems.- References.- Index.

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Book SynopsisThis textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams.After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential. Trade Review“Lynch has successfully captured this: I find this book to be uniquely successful in teaching a branch of mathematics together with computing while inspiring students to look at references and explorations beyond the text.” (Patrick Shipman, SIAM Review, Vol. 62 (2), 2020)“This book is meant as an upper level undergraduate or graduate text in dynamical systems. … this is an attractive text, one that I wish I had access to when I was learning dynamical systems, and one that I would be glad to teach from.” (John Starrett, MAA Reviews, July 28, 2019)Table of ContentsPreface.- A Tutorial Introduction to Python.- Differential Equations.- Planar Systems.- Interacting Species.- Limit Cycles.- Hamiltonian Systems, Lyapunov Functions, and Stability.- Bifurcation Theory.- Three-Dimensional Autonomous Systems and Chaos.- Poincaré Maps and Nonautonomous Systems in the Plane.- Local and Global Bifurcations.- The Second Part of Hilbert's Sixteenth Problem.- Delay Differential Equations.- Linear Discrete Dynamical Systems.- Nonlinear Discrete Dynamical Systems.- Complex Iterative Maps.- Electromagnectic Waves and Optical Resonators.- Fractals and Multifractals.- Image Processing with Python.- Chaos Control and Synchronization.- Neural Networks.- Binary Oscillator Computing.- Coursework and Examination-Type Questions.- Solutions to Exercises.- Index of Python Programs.- Index.

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Book SynopsisThis volume collects, in written form, eight plenary lectures and twenty-five selected contributions from invited and contributed lectures delivered at the International Workshop on Potential Theory 2004. The workshop was held at Shimane University, Matsue, Japan, from 23 to 28 August, 2004. The topic of the workshop was Potential Theory and its related fields. There were stimulus talks from classical potential theory to pluri-potential theory and probabilistic potential theory.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

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Book SynopsisThis volume is the proceedings of the 14th MSJ International Research Institute “Asymptotic Analysis and Singularity”, which was held at Sendai, Japan in July 2005. The proceedings contain survey papers and original research papers on nonlinear partial differential equations, dynamical systems, calculus of variations and mathematical physics.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North AmericaTable of ContentsWeak KAM pairs and Monge-Kantorovich duality by P. Bernard and B. Buffoni Hydrodynamic limit and nonlinear PDEs with singularities by T. Funaki PDE and differential geometry in study of motion of elastic wires by N. Koiso Evolution of stable phases in forward-backward parabolic equations by C. Mascia, A. Terracina, and A. Tesei Analogies between superconductors and liquid crystals: Nucleation and critical fields by X.-B. Pan Method of the blowup envelope and applications by T. Suzuki Numerical study and examples on singularities of solutions to anisotropic crystalline curvature flows of nonconvex polygonal curves by C. Hirota, T. Ishiwata, and S. Yazaki On certain one-dimensional elliptic systems under different growth conditions at respective infinities by M. Iida, K. Nakashima, and E. Yanagida Stability analysis for a stripe solution in the Gierer-Meinhardt system by K. Ikeda Perturbation of structures of radial solutions to elliptic equations by Y. Kabeya On the bifurcation structure of positive stationary solutions for a competition-diffusion system by Y. Kan-on Global solutions to a one-dimensional nonlinear parabolic system modeling colonial formation by chemotactic bacteria by K. P. P. Htoo, M. Mimura, and I. Takagi Existence of standing waves for the nonlinear Schrodinger equation with double power nonlinearity and harmonic potential by H. Kikuchi The Hamiltonian formalism in reaction-diffusion systems by M. Kuwamura Speed estimate for a periodic rotating wave in an undulating zone on the sphere by B. Lou On the dynamical system of a parabolic equation with non-local term by T. Miyasita A variational approach to self-similar solutions for semilinear heat equations by Y. Naito Determination of the limit sets of trajectories of the Gierer-Meinhardt system without diffusion by W.-M. Ni, K. Suzuki, and I. Takagi Asymptotic form of solutions of the Tadjbakhsh-Odeh variational problem by S. Okabe Global existence of a reaction-diffusion-advection system by K. Osaki Blowup solutions to some systems related to biology by T. Senba On the role of the source terms in an activator-inhibitor system proposed by Gierer and Meinhardt by K. Suzuki and I. Takagi Concentration phenomena in the conformal Brezis-Nirenberg problem by F. Takahashi Interfacial analysis to a chemotaxis model equation with growth in three dimension by T. Tsujikawa.

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Book SynopsisIn this invaluable book, macroscopic irreversible thermodynamics is presented in its realm and its splendor by appealing to the notion of internal variables of state. This applies to both fluids and solids with or without microstructures of mechanical or electromagnetic origin. This unmatched richness of essentially nonlinear behaviors is the result of the use of modern mathematical techniques such as convex analysis in a clear-cut framework which allows one to put under the umbrella of “irreversible thermodynamics” behaviors which until now have been commonly considered either not easily covered, or even impossible to incorporate into such a framework.The book is intended for all students and researchers whose main concern is the rational modeling of complex and/or new materials with physical and engineering applications, such as those accounting for coupled-field, hysteresis, fracture, nonlinear-diffusion, and phase-transformation phenomena.Table of ContentsIntroduction - a post-Duhemian thermodynamics; thermostatics and thermodynamics; various thermodynamics; thermodynamics with internal variables; applications - general framework; viscosity in complex fluids; viscoplasticity and plasticity; thermodynamics of fracture; non-equilibrium thermodynamics of electromagnetic materials; waves and reaction-diffusion systems (RDS).

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Book SynopsisThis textbook presents the most efficient analytical techniques in the local bifurcation theory of vector fields. It is centered on the theory of normal forms and its applications, including interaction with symmetries.The first part of the book reviews the center manifold reduction and introduces normal forms (with complete proofs). Basic bifurcations are studied together with bifurcations in the presence of symmetries. Special attention is given to examples with reversible vector fields, including the physical example given by the water waves. In this second edition, many problems with detailed solutions are added at the end of the first part (some systems being in infinite dimensions). The second part deals with the Couette-Taylor hydrodynamical stability problem, between concentric rotating cylinders. The spatial structure of various steady or unsteady solutions results directly from the analysis of the reduced system on a center manifold. In this part we also study bifurcations (simple here) from group orbits of solutions in an elementary way (avoiding heavy algebra). The third part analyzes bifurcations from time periodic solutions of autonomous vector fields. A normal form theory is developed, covering all cases, and emphasizing a partial Floquet reduction theory, which is applicable in infinite dimensions. Studies of period doubling as well as Arnold's resonance tongues are included in this part.Table of ContentsCentre manifolds normal forms, and bifurcations of vector; fields near critical points - unperturbed vector fields; perturbed vector fields; Couette-Taylor problem - formulation of the problem; Couette flow; bifurcations from Couette flow; bifurcations from Taylor vortex flow; centre manifolds, normal forms, and bifurcations of vector; fields near closed orbits - preliminaries; adaptation of Floquet thoery; unperturbed case; perturbed case.

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Book SynopsisSpirals, vortices, crystalline lattices, and other attractive patterns are prevalent in Nature. How do such beautiful patterns appear from the initial chaos? What universal dynamical rules are responsible for their formation? What is the dynamical origin of spatial disorder in nonequilibrium media? Based on the many visual experiments in physics, hydrodynamics, chemistry, and biology, this invaluable book answers those and related intriguing questions. The mathematical models presented for the dynamical theory of pattern formation are nonlinear partial differential equations. The corresponding theory is not so accessible to a wide audience. Consequently, the authors have made every attempt to synthesize long and complex mathematical calculations to exhibit the underlying physics. The book will be useful for final year undergraduates, but is primarily aimed at graduate students, postdoctoral fellows, and others interested in the puzzling phenomena of pattern formation.Table of ContentsPatterns - prelude to a dynamical description; linear stage of pattern formation; model equations; the Ginzburg-Landau equation; "crystal" formation; quasicrystals; breaking of order; localized patterns; spirals; patterns in oscillating soap films; patterns in colonies of micro-organisms; spatial disorder; patterns in chaotic media; epilogue - living matter and dynamic forms; a short guide to nonlinear dynamics; key experiments in pattern formation.

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Book SynopsisBifurcation and chaos has dominated research in nonlinear dynamics for over two decades, and numerous introductory and advanced books have been published on this subject. There remains, however, a dire need for a textbook which provides a pedagogically appealing yet rigorous mathematical bridge between these two disparate levels of exposition. This book has been written to serve that unfulfilled need.Following the footsteps of Poincaré, and the renowned Andronov school of nonlinear oscillations, this book focuses on the qualitative study of high-dimensional nonlinear dynamical systems. Many of the qualitative methods and tools presented in the book have been developed only recently and have not yet appeared in textbook form.In keeping with the self-contained nature of the book, all the topics are developed with introductory background and complete mathematical rigor. Generously illustrated and written at a high level of exposition, this invaluable book will appeal to both the beginner and the advanced student of nonlinear dynamics interested in learning a rigorous mathematical foundation of this fascinating subject.Table of ContentsStructurally Stable Systems; Bifurcations of Dynamical Systems; The Behavior of Dynamical Systems on Stability Boundaries of Equilibrium States; The Behavior of Dynamical Systems on Stability Boundaries of Periodic Trajectories; Local Bifurcations on the Route Over Stability Boundaries; Global Bifurcations at the Disappearance of a Saddle-Node Equilibrium States and Periodic Orbits; Bifurcations of Homoclinic Loops of Saddle Equilibrium States; Safe and Dangerous Boundaries.

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Book SynopsisAlthough the mathematical theory of nonlinear waves and solitons has made great progress, its applications to concrete physical problems are rather poor, especially when compared with the classical theory of linear dispersive waves and nonlinear fluid motion. The Whitham method, which describes the combining action of the dispersive and nonlinear effects as modulations of periodic waves, is not widely used by applied mathematicians and physicists, though it provides a direct and natural way to treat various problems in nonlinear wave theory. Therefore it is topical to describe recent developments of the Whitham theory in a clear and simple form suitable for applications in various branches of physics.This book develops the techniques of the theory of nonlinear periodic waves at elementary level and in great pedagogical detail. It provides an introduction to a Whitham's theory of modulation in a form suitable for applications. The exposition is based on a thorough analysis of representative examples taken from fluid mechanics, nonlinear optics and plasma physics rather than on the formulation and study of a mathematical theory. Much attention is paid to physical motivations of the mathematical methods developed in the book. The main applications considered include the theory of collisionless shock waves in dispersive systems and the nonlinear theory of soliton formation in modulationally unstable systems. Exercises are provided to amplify the discussion of important topics such as singular perturbation theory, Riemann invariants, the finite gap integration method, and Whitham equations and their solutions.Table of ContentsIntroduction and basic concepts; nonlinear wave equations in physics; Whitham theory of modulations; complete integrability of nonlinear wave equations; periodic solutions; dissipationless shock wave; nonlinear theory of modulational instability. Appendices: some formulas from the theory of elliptic functions; algebraic resolvents of fourth degree polynomials; solutions to exercises.

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Book SynopsisThis 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.Table of ContentsThe rhomboidal charged four body problem, F. Alfaro and E. Perez-Chavela; planetary rings with shepherds, L. Benet and T.H. Seligman; low Reynolds number swimming in two dimensions, A. Cherman et al; 2-dimensional invariant Tori for the spatial isosceles 3-body problem, M. Corbera and J. Llibre; the global flow for the synodical spatial Kepler problem, M.P. Dantas and J. Llibre; unbounded growth of energy in periodic perturbations of geodesic flows of the torus, A. Delshams et al; splitting and Meinikov potentials in Hamiltonian systems, A. Delshams and P. Gutierrez; infinity manifolds of cubic polynomial Hamiltonian vector fields with 2 degrees of freedom, M. Falconi et al; relativistic corrections to elementary Galilean dynamics and deformations of Poisson brackets, R. Flores-Espinoza and Y.M. Vorobjev; heteroclinic phenomena in the Sitnikov problem, A. Garcia and E. Perez-Chavela); doubly-symmetric periodic solutions of Hill's lunar problem, R.C. Howison and K.R. Meyer; on practical stability regions for the motion of a small particle close to the equilateral points of the real earth-moon system, A. Jorba; variational methods for quasi-periodic solutions of partial differential equations, R. de la Llave; the splitting of invariant Lagrangian submanifolds - geometry and dynamics, J.-P. Marco; cross-sections in the planar N-body problem, C. McCord; existence of an additional first integral and completeness of the flow for Hamiltonian vector fields, J. Mucino-Raymundo; simplification of perturbed Hamiltonians through Lie transformations, J. Palacian and P. Yanguas; linear stability in the 1 + N-Gon relative equilibrium, G.E. Roberts; analytic continuation of circular and elliptic Kepler motion to the general 3-body problem, J. Soler; the phase space of finite systems, K.B. Wolf et al.

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Book SynopsisThe responses of offshore structures are significantly affected by steep nonlinear waves, currents and wind, leading to phenomena such as springing and ringing of TLPs, slow drift yaw motion of FPSOs and large oscillations of Spar platforms due to vortex shedding. Research has brought about significant progress in this field over the past few decades and introduced us to increasingly involved concepts and their diverse applicability. Thus, an in-depth understanding of steep nonlinear waves and their effects on the responses of offshore structures is essential for safe and effective designs.This book deals with analyses of nonlinear problems encountered in the design of offshore structures, as well as those that are of immediate practical interest to ocean engineers and designers. It presents conclusions drawn from recent research pertinent to nonlinear waves and their effects on the responses of offshore structures. Theories, observations and analyses of laboratory and field experiments are expounded such that the nonlinear effects can be clearly visualized.Table of ContentsGaussian Processes and Wave Spectra; Propagation of Steady and Transient Large Linear Wave Groups; Nonlinear Wave Interaction; Laboratory Nonlinear Waves; Nonlinear Wave-Body Interaction; Volterra Input-Output Model; Universal Linear System Model for Nonlinear Loading, including Wave Impact; Numerical Wave Tank; Coupled Analysis of Nonlinear Responses of TLPs, FPSOs and Spar-Platform Systems; Fatigue Due to Non-Gaussian Forces.

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Book SynopsisThis 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.

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Book SynopsisThis 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.

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Book SynopsisIn recent years, enormous progress has been made on nonlinear dynamics particularly on chaos and complex phenomena. This unique volume presents the advances made in theory, analysis, numerical simulation and experimental realization, promising novel practical applications on various topics of current interest on chaos and related fields of nonlinear dynamics.Particularly, the focus is on the following topics: synchronization vs. chaotic phenomena, chaos and its control in engineering dynamical systems, fractal-based dynamics, uncertainty and unpredictability measures vs. chaos, Hamiltonian systems and systems with time delay, local/global stability, bifurcations and their control, applications of machine learning to chaos, nonlinear vibrations of lumped mass mechanical/mechatronic systems (rigid body and coupled oscillator dynamics) governed by ODEs and continuous structural members (beams, plates, shells) vibrations governed by PDEs, patterns formation, chaos in micro- and nano-mechanical systems, chaotic reduced-order models, energy absorption/harvesting from chaotic, chaos vs. resonance phenomena, chaos exhibited by discontinuous systems, chaos in lab experiments.The present volume forms an invaluable source on recent trends in chaotic and complex dynamics for any researcher and newcomers to the field of nonlinear dynamics.

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Book SynopsisThe book is devoted to the mathematical foundations of nonextensive statistical mechanics. This is the first book containing the systematic presentation of the mathematical theory and concepts related to nonextensive statistical mechanics, a current generalization of Boltzmann-Gibbs statistical mechanics introduced in 1988 by one of the authors and based on a nonadditive entropic functional extending the usual Boltzmann-Gibbs-von Neumann-Shannon entropy. Main mathematical tools like the q-exponential function, q-Gaussian distribution, q-Fourier transform, q-central limit theorems, and other related objects are discussed rigorously with detailed mathematical rational. The book also contains recent results obtained in this direction and challenging open problems. Each chapter is accompanied with additional useful notes including the history of development and related bibliographies for further reading.

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Book SynopsisA recent development is the discovery that simple systems of equations can have chaotic solutions in which small changes in initial conditions have a large effect on the outcome, rendering the corresponding experiments effectively irreproducible and unpredictable. An earlier book in this sequence, Elegant Chaos: Algebraically Simple Chaotic Flows provided several hundred examples of such systems, nearly all of which are purely mathematical without any obvious connection with actual physical processes and with very limited discussion and analysis.In this book, we focus on a much smaller subset of such models, chosen because they simulate some common or important physical phenomenon, usually involving the motion of a limited number of point-like particles, and we discuss these models in much greater detail. As with the earlier book, the chosen models are the mathematically simplest formulations that exhibit the phenomena of interest, and thus they are what we consider 'elegant.'Elegant models, stripped of unnecessary detail while maximizing clarity, beauty, and simplicity, occupy common ground bordering both real-world modeling and aesthetic mathematical analyses. A computational search led one of us (JCS) to the same set of differential equations previously used by the other (WGH) to connect the classical dynamics of Newton and Hamilton to macroscopic thermodynamics. This joint book displays and explores dozens of such relatively simple models meeting the criteria of elegance, taste, and beauty in structure, style, and consequence.This book should be of interest to students and researchers who enjoy simulating and studying complex particle motions with unusual dynamical behaviors. The book assumes only an elementary knowledge of calculus. The systems are initial-value iterated maps and ordinary differential equations but they must be solved numerically. Thus for readers a formal differential equations course is not at all necessary, of little value and limited use.

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Book SynopsisThe book is devoted to the problems of modeling physical systems and fields using the tools and capabilities of the 'Mathematica' software package. In the process of teaching classical courses in mechanics and mathematical physics, one often has to overcome significant difficulties associated with the cumbersomeness of the mathematical apparatus, which more than once distracts from the essence of the problems under consideration. The use of the 'Mathematica' package, which has a rich set of analytical and graphic tools, makes the presentation of classic issues related to modeling and interpretation of physical processes much more transparent. This package enables the visualization of both analytical solutions of nonlinear differential equations and solutions obtained in the form of infinite series or special functions.The textbook consists of two parts that can be studied independently of each other. The first part deals with the issues of nonlinear mechanics and the theory of oscillations. The second part covers linear problems of classical mathematical physics and nonlinear evolution models describing, inter alia, transport phenomena and propagation of waves. The book contains the codes of programs written in the 'Mathematica' package environment. Supplementary materials of programs illustrating and often complementing the presented material are available on the publisher's website.

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Book SynopsisIn the early 1980s, the late luminary Tito Arecchi was the first to highlight the existence of chaos in a laser model. Since then, along with several colleagues, he developed many important lines of research in this field, such as generalized multistability, laser with injected signal, laser with delayed feedback and the worldwide accepted classification of lasers of A, B and C, depending on their typical relaxation rates. Later, chaos control and synchronization were investigated in lasers and other systems, providing innovative schemes. Very recently, in his last contribution to laser physics, the model of the laser with feedback demonstrating its universal features was revisited.This book aims to present the research activity of Prof. Arecchi and his colleagues in the domain of nonlinear dynamics of lasers, since his seminal works of 1982 till the latest. Also included is our last contribution on jerk dynamics of laser's minimal universal model and a brief history of the discovery of laser where the reader will discover or rediscover many anecdotes about it.

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Book SynopsisThe topic of nonlinear systems is fundamental to the study of systems engineering. So extensive investigations have been carried out by both the nonlinear control and nonlinear dynamics communities, but the focus can be different — on controllers design and dynamics analysis, respectively. The last two decades have witnessed the gradual merging of control theory and dynamics analysis, but not yet to the extent of controlling nonlinear dynamics such as bifurcations and chaos. This monograph is an attempt to fill that gap while presenting a rather comprehensive coverage of the fundamental nonlinear systems theory in a self-contained and approachable manner.This introductory treatise is written for self-study and, in particular, as an elementary textbook that can be taught in a one-semester course to advanced undergraduates or entrance level graduates with curricula focusing on nonlinear systems, both on control theory and dynamics analysis.

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Book SynopsisOrdinary differential equations is a standard course in the undergraduate mathematics curriculum that usually comes after the first university calculus and linear algebra courses taken by a mathematics major. Such courses may also typically be attended by undergraduates from other areas of physical and social sciences, and engineering. The content of such a course has remained fairly static over time, despite the expansion of the topic into other disciplines as a result of the dynamical systems point of view.This core undergraduate course updated from the dynamical systems perspective can easily be covered in one semester, with room for projects or more advanced topics tailored to the interests of the students.

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Book SynopsisOrdinary differential equations is a standard course in the undergraduate mathematics curriculum that usually comes after the first university calculus and linear algebra courses taken by a mathematics major. Such courses may also typically be attended by undergraduates from other areas of physical and social sciences, and engineering. The content of such a course has remained fairly static over time, despite the expansion of the topic into other disciplines as a result of the dynamical systems point of view.This core undergraduate course updated from the dynamical systems perspective can easily be covered in one semester, with room for projects or more advanced topics tailored to the interests of the students.

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Book SynopsisThe author of this book presents conceptual and experimental evidence showing that Heisenberg's uncertainty relations are not valid in all cases. Furthermore, he derives a more general set of uncertainty relations. The new relations result from the replacement of the Fourier nonlocal and nontemporal paradigm by wavelet local analysis. These results lead to a coherent and beautiful causal synthesis unifying quantum and classical physics.Table of ContentsComplementarity Principle and the Nonlocal Fourier Analysis; New Generation of Microscopes; Beyond Heisenberg's Uncertainty Relations; The Meaning of the Wave Function; Interference Without Superposition of the Wave Packets.

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Book SynopsisThe shared platform of the articles collected in this volume is used to advocate a dynamical systems approach to cognition. It is argued that recent developments in cognitive science towards an account of embodiment, together with the general approach of complexity theory and dynamics, have a major impact on behavioral and cognitive science. The book points out that there are two domains that follow naturally from the stance of embodiment: first, coordination dynamics is an established empirical paradigm that is best able to aid the approach; second, the obvious goal-directedness of intelligent action (i.e., intentionality) is nicely addressed in the framework of the dynamical synergetic approach.Table of ContentsIntelligent Behavior - A Synergetic View (H Haken); Cognitive Coordination Dynamics (S Kelso); Grounded in the World: Developmental Origins of the Embodied Mind (Revised Reprint) (E Thelen); What is Coordinated in Bimanual Coordination? (F Mechsner & W Prinz); Cognition in Action: The Interplay of Attention and Bimanual Coordination Dynamics (J J Temprado et a.); A Synergetic Approach to Describe the Stability and Variability of Motor Behaviour (K Witte et al.); The Role of Synchronization in Perception-Action (T-C Chan et al.); A Mean-Field Approach to Self-Organization in Spatially Extended Perception-Action and Psychological Systems (T Frank & P J Beek); Efficiency Aspects of Self-Organizing Systems (W Tschacher et al.); The Embodiment of Intentionality (S Jordan); Cognitive Science, Representations and Dynamical Systems Theory (W F G Haselager); Self-Steered Self-Organization (F Keijzer); SIRN (Synergetic Inter-Representation Networks), Artifacts and Snow's Two Cultures (J Fortugali); Brain Dynamics: Methodological Issues and Applications in Psychiatic and Neurologic Diseases (L Fezard); Dynamical Systems Theory: Applications to Pedagogy (J L Abraham).

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Book SynopsisNonlinear dynamics of complex processes is an active research field with large numbers of publications in basic research, and broad applications from diverse fields of science. Nonlinear dynamics as manifested by deterministic and stochastic evolution models of complex behavior has entered statistical physics, physical chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics, etc. This field of research has induced a new terminology in science connected with new questions, problems, solutions and methods. New scenarios have emerged for spatio-temporal structures in dynamical systems far from equilibrium. Their analysis and possible control are intriguing and challenging aspects of the current research.The duality of fundamental and applied research is a focal point of its main attractivity and fascination. Basic topics and foundations are always linked to concrete and precise examples. Models and measurements of complex nonlinear processes evoke and provoke new fundamental questions that diversify and broaden the mathematical concepts and tools. In return, new mathematical approaches to modeling and analysis enlarge the scope and efficiency of applied research.Table of ContentsNoise-Induced Effects in Excitable Systems with Local and Global Coupling (X R Sailer et al.); Spiral Wave Dynamics: Reaction and Diffusion versus Kinematics (B Fiedler et al.); Pattern Formation in Semiconductors Under the Influence of Time-Delayed Feedback Control and Noise (E Scholl et al.); Trapping of Phase Fronts and Twisted Spirals in Periodically Forced Oscillatory Media (O Rudzick & A S Mikhailov); Unified Approach to Feedback-Mediated Control of Spiral Waves in Excitable Media (V S Zykov & H Engel); Building Oscillations Bottom Up: Elemental Time Scales of intracellular Calcium Dynamics (R Thul & M Falcke); Synchronization of Complex Systems: Analysis and Control (M Rosenblum & A Pikovsky); Predator-Prey Oscillations, Synchronization and Pattern Formation in Ecological Systems (B Blasius & R Tonjes); and other papers.

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Book SynopsisPiecewise constant systems exist in widely expanded areas such as engineering, physics, and mathematics. Extraordinary and complex characteristics of piecewise constant systems have been reported in recent years. This book provides the methodologies for analyzing and assessing nonlinear piecewise constant systems on a theoretically and practically sound basis. Recently developed approaches for theoretically analyzing and numerically solving the nonlinear piecewise constant dynamic systems are reviewed. A new greatest integer argument with a piecewise constant function is utilized for nonlinear dynamic analyses and for establishing a novel criterion in diagnosing irregular and chaotic solutions from the regular solutions of a nonlinear dynamic system. The newly established piecewise constantization methodology and its implementation in analytically solving for nonlinear dynamic problems are also presented.Table of ContentsFundamentals of Conventional and Piecewise Constant Systems; Preliminary Theorems and Techniques for Analysis of Nonlinear Piecewise Constant Systems; Piecewise Constant Dynamical Systems and Their Behavior; Analytical and Semi-Analytical Solution Development with Piecewise Constant Arguments; Numerical and Improved Semi-Analytical Approaches Implementing Piecewise Constant Arguments; Application of P-T Method on Multi-Degree-of-Freedom Nonlinear Dynamic Systems; Periodicity-Ratio and Its Application in Diagnosing Irregularities of Nonlinear Systems.

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Book SynopsisNew Edition: The Nonlinear Workbook (6th Edition)The study of nonlinear dynamical systems has advanced tremendously in the last 20 years, making a big impact on science and technology. This book provides all the techniques and methods used in nonlinear dynamics. The concepts and underlying mathematics are discussed in detail.The numerical and symbolic methods are implemented in C++, SymbolicC++ and Java. Object-oriented techniques are also applied. The book contains more than 150 ready-to-run programs.The text has also been designed for a one-year course at both the junior and senior levels in nonlinear dynamics. The topics discussed in the book are part of e-learning and distance learning courses conducted by the International School for Scientific Computing, University of Johannesburg.Table of ContentsNonlinear and Chaotic Maps; Time Series Analysis; Autonomous Systems in the Plane; Nonlinear Hamilton Systems; Nonlinear Dissipative Systems; Nonlinear Driven Systems; Controlling of Chaos; Synchronization of Chaos; Fractals; Cellular Automata; Solving Differential Equations; Neural Networks; Genetic Algorithms; Gene Expression Programming; Optimization; Discrete Wavelets; Discrete Hidden Markov Processes; Fuzzy Sets and Fuzzy Logic.

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Book SynopsisThis unique book aims to treat a class of nonlinear waves that are reflected from the boundaries of media of finite extent. It involves both standing (unforced) waves and resonant oscillations due to external periodic forcing. The waves are both hyperbolic and dispersive. To achieve this aim, the book develops the necessary understanding of linear waves and the mathematical techniques of nonlinear waves before dealing with nonlinear waves in bounded media. The examples used come mainly from gas dynamics, water waves and viscoelastic waves.

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Book SynopsisThis 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.

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Book SynopsisThis 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.

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Book SynopsisThis elegant book presents a rigorous introduction to the theory of nonlinear mechanics and chaos. It turns out that many simple mechanical systems suffer from a peculiar malady. They are deterministic in the sense that their motion can be described with partial differential equations, but these equations have no proper solutions and the behavior they describe can be wildly unpredictable. This is implicit in Newtonian physics, and although it was analyzed in the pioneering work of Poincare in the 19th century, its full significance has only been realized since the advent of modern computing. This book follows this development in the context of classical mechanics as it is usually taught in most graduate programs in physics. It starts with the seminal work of Laplace, Hamilton, and Liouville in the early 19th century and shows how their formulation of mechanics inevitably leads to systems that cannot be 'solved' in the usual sense of the word. It then discusses perturbation theory which, rather than providing approximate solutions, fails catastrophically due to the problem of small denominators. It then goes on to describe chaotic motion using the tools of discrete maps and Poincare sections. This leads to the two great landmarks of chaos theory, the Poincare-Birkhoff theorem and the so-called KAM theorem, one of the signal results in modern mathematics. The book concludes with an appendix discussing the relevance of the KAM theorem to the ergodic hypothesis and the second law of thermodynamics.Lectures on Nonlinear Mechanics and Chaos Theory is written in the easy conversational style of a great teacher. It features numerous computer-drawn figures illustrating the behavior of nonlinear systems. It also contains homework exercises and a selection of more detailed computational projects. The book will be valuable to students and faculty in physics, mathematics, and engineering.See Press Release: Problems in mechanics open the door to the orderly world of chaosTable of ContentsLagrangian Dynamics; Noether's Theorem; Hamiltonian Formulation; Hamilton's Principle Function; Hamilton's Characteristic Function; Action-Angle Variables; Abstract Transformation Theory; Poisson Brackets; Liouville's Theorem; Perturbation Theory; The Henon-Heiles Oscillator; Discrete Maps; Lyapunov Exponents; The Poincare-Birkhoff Theorem; The KAM Theorem; Ergodic Hypothesis; Measure Theory;

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Book SynopsisThis elegant book presents a rigorous introduction to the theory of nonlinear mechanics and chaos. It turns out that many simple mechanical systems suffer from a peculiar malady. They are deterministic in the sense that their motion can be described with partial differential equations, but these equations have no proper solutions and the behavior they describe can be wildly unpredictable. This is implicit in Newtonian physics, and although it was analyzed in the pioneering work of Poincare in the 19th century, its full significance has only been realized since the advent of modern computing. This book follows this development in the context of classical mechanics as it is usually taught in most graduate programs in physics. It starts with the seminal work of Laplace, Hamilton, and Liouville in the early 19th century and shows how their formulation of mechanics inevitably leads to systems that cannot be 'solved' in the usual sense of the word. It then discusses perturbation theory which, rather than providing approximate solutions, fails catastrophically due to the problem of small denominators. It then goes on to describe chaotic motion using the tools of discrete maps and Poincare sections. This leads to the two great landmarks of chaos theory, the Poincare-Birkhoff theorem and the so-called KAM theorem, one of the signal results in modern mathematics. The book concludes with an appendix discussing the relevance of the KAM theorem to the ergodic hypothesis and the second law of thermodynamics.Lectures on Nonlinear Mechanics and Chaos Theory is written in the easy conversational style of a great teacher. It features numerous computer-drawn figures illustrating the behavior of nonlinear systems. It also contains homework exercises and a selection of more detailed computational projects. The book will be valuable to students and faculty in physics, mathematics, and engineering.See Press Release: Problems in mechanics open the door to the orderly world of chaosTable of ContentsLagrangian Dynamics; Noether's Theorem; Hamiltonian Formulation; Hamilton's Principle Function; Hamilton's Characteristic Function; Action-Angle Variables; Abstract Transformation Theory; Poisson Brackets; Liouville's Theorem; Perturbation Theory; The Henon-Heiles Oscillator; Discrete Maps; Lyapunov Exponents; The Poincare-Birkhoff Theorem; The KAM Theorem; Ergodic Hypothesis; Measure Theory;

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Book SynopsisThe main goal is to offer to readers a panorama of recent progress in nonlinear physics, complexity and transport with attractive chapters readable by a broad audience. It allows to gain an insight into these active fields of research and notably promotes the interdisciplinary studies from mathematics to experimental physics. To reach this aim, the book collects a selection of contributions to the third edition of the CCT conference (Marseilles, 1-5 June 2015).

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Book SynopsisOur 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.

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Book SynopsisOur 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.

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Book SynopsisHigh technology industries are in desperate need for adequate tools to assess the validity of simulations produced by ever faster computers for perennial unstable problems. In order to meet these industrial expectations, applied mathematicians are facing a formidable challenge summarized by these words — nonlinearity and coupling. This book is unique as it proposes truly original solutions: (1) Using hypercomputation in quadratic algebras, as opposed to the traditional use of linear vector spaces in the 20th century; (2) complementing the classical linear logic by the complex logic which expresses the creative potential of the complex plane.The book illustrates how qualitative computing has been the driving force behind the evolution of mathematics since Pythagoras presented the first incompleteness result about the irrationality of √2. The celebrated results of Gödel and Turing are but modern versions of the same idea: the classical logic of Aristotle is too limited to capture the dynamics of nonlinear computation. Mathematics provides us with the missing tool, the organic logic, which is aptly tailored to model the dynamics of nonlinearity. This logic will be the core of the “Mathematics for Life” to be developed during this century.Table of ContentsIntroduction to Qualitative Computing; Hypercomputation in Dickson Algebras; Scales of Complexity and Linear Reachability; Singular Values for the Multiplication Maps; Computation Beyond Classical Logic; Complexification of the Arithmetic; Homotopic Deviation in Linear Algebra; The Discrete and the Continuous; Arithmetic in the Alternative Dickson Division Algebras; The Real and the Complex.

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