Chaos theory Books

Book SynopsisThis book is devoted to the history of chaos theory, from celestial mechanics (three-body problem) to electronics and meteorology. Many illustrative examples of chaotic behaviors exist in various contexts found in nature (chemistry, astrophysics, biomedicine). This book includes the most popular systems from chaos theory (Lorenz, Rössler, van der Pol, Duffing, logistic map, Lozi map, Hénon map etc.) and introduces many other systems, some of them very rarely discussed in textbooks as well as in scientific papers. The contents are formulated with an original approach as compared to other books on chaos theory.

Read more less

Book Synopsis'This book offers one of the few places where a collection of results from the literature can be found … The book has an extensive bibliography … It is very nice to have the compendium of results that is presented here.'zbMATHA mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.

Read more less

Book SynopsisWhat do economic chaos and uncertainties mean in rational or irrational economic theories? How do simple deterministic interactions among a few variables lead to unpredictable complex phenomena? Why is complexity of economies causing so many conflicts and confusions worldwide?This book provides a comprehensive introduction to recent developments of complexity theory in economics. It presents different models based on well-accepted economic mechanisms such as the Solow model, Ramsey model, and Lucas model. It is focused on presenting complex behaviors, such as business cycles, aperiodic motion, bifurcations, catastrophes, chaos, and hidden attractors, in basic economic models with nonlinear behavior. It shows how complex nonlinear phenomena are identified from various economic mechanisms and theories. These models demonstrate that the traditional or dominant economic views on evolution of, for instance, capitalism market, free competition, or Keynesian economics, are not generally valid. Markets are unpredictable and nobody knows with certainty the consequences of policies or other external factors in economic systems with simple interactions.

Read more less

Book SynopsisThis book offers a fun and enriching introduction to chaos theory, fractals and dynamical systems, and on the applications of fractals to computer generated graphics and image compression. Introduction to Chaos, Fractals and Dynamical Systems particularly focuses on natural and human phenomenon that can be modeled as fractals, using simple examples to explain the theory of chaos and how it affects all of us. Then, using straightforward mathematic and intuitive descriptions, computer generated graphics and photographs of natural scenes are used to illustrate the beauty of fractals and their importance in our world. Finally, the concept of Dynamical Systems, that is, time-dependent systems, the foundation of Chaos and Fractal, is introduced. Everyday examples are again used to illustrate concepts, and the importance of understanding how these vital systems affect our lives. Throughout the fascinating history of the evolution of chaos theory, fractals and dynamical systems is presented, along with brief introductions to the scientists, mathematicians and engineers who created this knowledge.Introduction to Chaos, Fractals and Dynamical Systems contains ample mathematical definitions, representations, discussions and exercises, so that this book can be used as primary or secondary source in home schooling environments.The book is suitable for homeschooling as a focused course on the subject matter or as a classroom supplement for a variety of courses at the late junior high or early high-school level. For example, in addition to a standalone course on Chaos, Fractals and Dynamical Systems (or similar title), this book could be used with the following courses:The text can also be used in conjunction with mathematics courses for undergraduates for non-science majors. The book can also be used for informal and lively family study and discussion.For each chapter, exercises and things to do are included. These activities range from simple computational tasks to more elaborate computer projects, related activities, biographical research and writing assignments.

Read more less

Book SynopsisThis book offers a fun and enriching introduction to chaos theory, fractals and dynamical systems, and on the applications of fractals to computer generated graphics and image compression. Introduction to Chaos, Fractals and Dynamical Systems particularly focuses on natural and human phenomenon that can be modeled as fractals, using simple examples to explain the theory of chaos and how it affects all of us. Then, using straightforward mathematic and intuitive descriptions, computer generated graphics and photographs of natural scenes are used to illustrate the beauty of fractals and their importance in our world. Finally, the concept of Dynamical Systems, that is, time-dependent systems, the foundation of Chaos and Fractal, is introduced. Everyday examples are again used to illustrate concepts, and the importance of understanding how these vital systems affect our lives. Throughout the fascinating history of the evolution of chaos theory, fractals and dynamical systems is presented, along with brief introductions to the scientists, mathematicians and engineers who created this knowledge.Introduction to Chaos, Fractals and Dynamical Systems contains ample mathematical definitions, representations, discussions and exercises, so that this book can be used as primary or secondary source in home schooling environments.The book is suitable for homeschooling as a focused course on the subject matter or as a classroom supplement for a variety of courses at the late junior high or early high-school level. For example, in addition to a standalone course on Chaos, Fractals and Dynamical Systems (or similar title), this book could be used with the following courses:The text can also be used in conjunction with mathematics courses for undergraduates for non-science majors. The book can also be used for informal and lively family study and discussion.For each chapter, exercises and things to do are included. These activities range from simple computational tasks to more elaborate computer projects, related activities, biographical research and writing assignments.

Read more less

Book SynopsisWhat kind of dynamics is a piecewise linear system able to display? How may they generate heteroclinic chaos? How can the coexistence of attractors be designed and characterized? Is it necessary to have equilibrium points to generate chaotic behavior? Chaos theory and complex systems are interesting and evolving topics whose investigation from a theoretical and practical point of view constantly leads to arising questions. Interesting behaviors can be observed in self-excited attractors, hidden attractors and non-self-excited attractors.This book presents some fundamentals of linear system theory and recent approaches to design the three classes of chaotic attractors in piecewise linear systems. Each chapter presents a brief description and basic concepts to provide an overview of linear systems theory; chaos and multistability in integer linear systems; hidden and non-self-excited attractors; and fractional approaches. They also provide example systems to illustrate the concepts and design methods introduced. Some current topics under investigation are addressed from an integer order perspective to make the connection with the fractional order counterpart.This textbook provides a comprehensive introduction, methodologies, and analysis tools to study chaotic piecewise linear systems and will be suitable for undergraduate or graduate students interested in the field of chaos and complex dynamics.

Read more less

Book SynopsisCellular automata are a class of spatially and temporally discrete mathematical systems characterized by local interaction and synchronous dynamical evolution. Introduced by the mathematician John von Neumann in the 1950s as simple models of biological self-reproduction, they are prototypical models for complex systems and processes consisting of a large number of simple, homogeneous, locally interacting components. Cellular automata have been the focus of great attention over the years because of their ability to generate a rich spectrum of very complex patterns of behavior out of sets of relatively simple underlying rules. Moreover, they appear to capture many essential features of complex self-organizing cooperative behavior observed in real systems.This book provides a summary of the basic properties of cellular automata, and explores in depth many important cellular-automata-related research areas, including artificial life, chaos, emergence, fractals, nonlinear dynamics, and self-organization. It also presents a broad review of the speculative proposition that cellular automata may eventually prove to be theoretical harbingers of a fundamentally new information-based, discrete physics. Designed to be accessible at the junior/senior undergraduate level and above, the book will be of interest to all students, researchers, and professionals wanting to learn about order, chaos, and the emergence of complexity. It contains an extensive bibliography and provides a listing of cellular automata resources available on the World Wide Web.Table of Contents* Introduction: Preliminary Musings * Formalism * Phenomenological Studies of Generic CA * Dynamical Systems Theory Approach * Analytic Approach * Cellular Automata and Language Theory * Probabilistic CA * Generalized Models * CA Models of Fluid Dynamics * Neural Networks * Artificial-Life * Is Nature, Underneath It All, a CA?

Read more less

Book SynopsisThis book is the first monograph devoted exclusively to strange nonchaotic attractors (SNA), recently discovered objects with a special kind of dynamical behavior between order and chaos in dissipative nonlinear systems under quasiperiodic driving. A historical review of the discovery and study of SNA, mathematical and physically-motivated examples, and a review of known experimental studies of SNA are presented. The main focus is on the theoretical analysis of strange nonchaotic behavior by means of different tools of nonlinear dynamics and statistical physics (bifurcation analysis, Lyapunov exponents, correlations and spectra, renormalization group). The relations of the subject to other fields of physics such as quantum chaos and solid state physics are also discussed.Trade Review"This book is written very precisely with a lot of illustrations. It is indispensable for physicists and mathematicians interested in the above stated dynamical systems and their SNA."Zentralblatt MATH"This is the first attempt to gather up all the work on this topic and present it in a unified way ... the book does an excellent job in outlining the main examples, results and connections to the literature and should be an invaluable tool for anyone wishing to find out about SNAs."Mathematical Reviews

Read more less

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).

Read more less

Book SynopsisChaos: from simple models to complex systems aims to guide science and engineering students through chaos and nonlinear dynamics from classical examples to the most recent fields of research. The first part, intended for undergraduate and graduate students, is a gentle and self-contained introduction to the concepts and main tools for the characterization of deterministic chaotic systems, with emphasis to statistical approaches. The second part can be used as a reference by researchers as it focuses on more advanced topics including the characterization of chaos with tools of information theory and applications encompassing fluid and celestial mechanics, chemistry and biology.The book is novel in devoting attention to a few topics often overlooked in introductory textbooks and which are usually found only in advanced surveys such as: information and algorithmic complexity theory applied to chaos and generalization of Lyapunov exponents to account for spatiotemporal and non-infinitesimal perturbations.The selection of topics, numerous illustrations, exercises and proposals for computer experiments make the book ideal for both introductory and advanced courses.

Read more less

Book SynopsisA periodically forced mathematical pendulum is one of the typical and popular nonlinear oscillators that possess complex and rich dynamical behaviors. Although the pendulum is one of the simplest nonlinear oscillators, yet, until now, we are still not able to undertake a systematical study of periodic motions to chaos in such a simplest system due to lack of suitable mathematical methods and computational tools. To understand periodic motions and chaos in the periodically forced pendulum, the perturbation method has been adopted. One could use the Taylor series to expend the sinusoidal function to the polynomial nonlinear terms, followed by traditional perturbation methods to obtain the periodic motions of the approximated differential system.This book discusses Hamiltonian chaos and periodic motions to chaos in pendulums. This book first detects and discovers chaos in resonant layers and bifurcation trees of periodic motions to chaos in pendulum in the comprehensive fashion, which is a base to understand the behaviors of nonlinear dynamical systems, as a results of Hamiltonian chaos in the resonant layers and bifurcation trees of periodic motions to chaos. The bifurcation trees of travelable and non-travelable periodic motions to chaos will be presented through the periodically forced pendulum.

Read more less

Book Synopsis'This book offers one of the few places where a collection of results from the literature can be found … The book has an extensive bibliography … It is very nice to have the compendium of results that is presented here.'zbMATHA mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.

Read more less

Book SynopsisThis volume includes the best papers presented at the CHAOS 2008 International Conference on Chaotic Modeling, Simulation and Applications. It provides a valuable collection of new ideas, methods, and techniques in the field of nonlinear dynamics, chaos, fractals and their applications in general science and in engineering sciences.It touches on many fields such as chaos, dynamical systems, nonlinear systems, fractals and chaotic attractors. It also covers mechanics, hydrofluid dynamics, chaos in meteorology and cosmology, Hamiltonian and quantum chaos, chaos in biology and genetics, chaotic control, and chaos in economy and markets, and chaotic simulations; thus, containing cutting-edge interdisciplinary research with high-interest applications.These contributions present new solutions by analyzing the relevant data and through the use of recent advances in different fields, especially in chaotic simulation methods and techniques.Table of ContentsChaos and Dynamical Systems; Nonlinear Systems; Fractals; Chaotic Attractors; Mechanics; Hydrofluid Dynamics; Chaotic Advection; Chaos in Meteorology and Cosmology; Bifurcation; Hamiltonian and Quantum Chaos; Plasma Physics; Chaos in Biology and Genetics; Chaos in Medicine and Physiology; Chaotic Control; Time Series Analysis and Forecasting Chaotic Systems; Chaos in Economy and Markets; Traffic Flow; Chaotic Simulations.

Read more less

Book SynopsisChaos: from simple models to complex systems aims to guide science and engineering students through chaos and nonlinear dynamics from classical examples to the most recent fields of research. The first part, intended for undergraduate and graduate students, is a gentle and self-contained introduction to the concepts and main tools for the characterization of deterministic chaotic systems, with emphasis to statistical approaches. The second part can be used as a reference by researchers as it focuses on more advanced topics including the characterization of chaos with tools of information theory and applications encompassing fluid and celestial mechanics, chemistry and biology.The book is novel in devoting attention to a few topics often overlooked in introductory textbooks and which are usually found only in advanced surveys such as: information and algorithmic complexity theory applied to chaos and generalization of Lyapunov exponents to account for spatiotemporal and non-infinitesimal perturbations.The selection of topics, numerous illustrations, exercises and proposals for computer experiments make the book ideal for both introductory and advanced courses.Table of ContentsFirst Encounter with Chaos; The Language of Dynamical Systems; Examples of Chaotic Behaviors; Probabilistic Approach to Chaos; Characterization of Chaotic Dynamical Systems; From Order to Chaos in Dissipative Systems; Chaos in Hamiltonian Systems; Chaos and Information Theory; Coarse-Grained Information and Large Scale Predictability; Chaos, Numerical Computations and Experiments; Chaos in Few-Degrees of Freedom Systems; Spatiotemporal Chaos; Turbulence as a Dynamical Systems Problem; Chaos and Statistical Mechanics: Fermi-Pasta-Ulam a Case Study.

Read more less

Book SynopsisThis review volume consists an indispensable collection of research papers chronicling the recent progress in controlling chaos. Here, new theoretical ideas, as experimental implementations of controlling chaos, are included, while the applications contained in this volume can be referred to as turbulent magnetized plasmas, chaotic neural networks, modeling city traffic and models of interest in celestial mechanics.Recent Progress in Controlling Chaos provides an excellent broad overview of the subject matter, and will be especially useful for graduate students, researchers and scientists working in the areas of nonlinear dynamics, chaos and complex systems. The authors, world-renowned scientists and prominent experts in the field of controlling chaos, will offer readers through their research works, a fascinating insight into the state-of-the-art technology used in the progress in key techniques and concepts in the field of control.Table of ContentsReduction of Chaotic Transport in Plasmas; Controlling Chaos in Chaotic Neural Networks; Adaptive Feedback Control; Feedback Anti-Control of Chaos; Delayed Feedback Control; Phase Control of Chaotic Systems; Controlling Chaos in Mechanical Systems; Clipping Chaos to Cycles; Controlling City Traffic; Controlling Bursting Neurons; Partial Control of Chaos; Pulsive Feedback Control; Chaos Control; Chaos Stabilization in the Three Body Problem; Stabilizing Periodic Orbits in Satellites.

Read more less

Book SynopsisReal-world systems that involve some non-smooth change are often well-modeled by piecewise-smooth systems. However there still remain many gaps in the mathematical theory of such systems. This doctoral thesis presents new results regarding bifurcations of piecewise-smooth, continuous, autonomous systems of ordinary differential equations and maps. Various codimension-two, discontinuity induced bifurcations are unfolded in a rigorous manner. Several of these unfoldings are applied to a mathematical model of the growth of Saccharomyces cerevisiae (a common yeast). The nature of resonance near border-collision bifurcations is described; in particular, the curious geometry of resonance tongues in piecewise-smooth continuous maps is explained in detail.Neimark-Sacker-like border-collision bifurcations are both numerically and theoretically investigated. A comprehensive background section is conveniently provided for those with little or no experience in piecewise-smooth systems.Table of ContentsFundamentals of Piecewise-Smooth, Continuous Systems; Discontinuous Bifurcations in Planar Systems; Codimension-Two, Discontinuous Bifurcations; The Growth of Saccharomyces cerevisiae; Codimension-Two, Border-Collision Bifurcations; Periodic Solutions and Resonance Tongues; Neimark-Sacker-Like Bifurcations;

Read more less

Book SynopsisThis book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hénon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters.Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hénon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincaré map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincaré mapping in addition to other analytical methods.

Read more less

Book SynopsisThe book is a compilation of selected papers from the conference on Physics and Control 2009, presenting a unified perspective underlying the thematics and strategies related to the control of physical systems with emerging applications in physics, engineering, chemistry, biology and other natural sciences. The selected papers reflect the state-of-the-art of the more advanced theoretical and practical studies in the field of control of complex systems. The contributions provide a comprehensive view on some selected topics of particular importance at the disciplinary borderline between Physics and Control.Table of ContentsModelling and Control of Coupled Stochastic Oscillators; Multistability in Natural and Laboratory-Scale Nonlinear Systems; Linear and Matritial Algebra, Open Problems Related to Control Theory; Localization of Oscillations in Dynamical Systems; Microfluidics: Theory, Methods and Applications; Mathematical Modelling of Dynamic Systems for Volcano Physics; Geometric Control for Quantum and Classical Models; Control Problems for Dynamical Systems Under Uncertainty and Conflict; Physics and Control in Fusion Plasma Devices; Modelling and Optimization of Beam and Plasma Dynamics.

Read more less

Book SynopsisPendulum is the simplest nonlinear system, which, however, provides the means for the description of different phenomena in Nature that occur in physics, chemistry, biology, medicine, communications, economics and sociology. The chaotic behavior of pendulum is usually associated with the random force acting on a pendulum (Brownian motion). Another type of chaotic motion (deterministic chaos) occurs in nonlinear systems with only few degrees of freedom. This book presents a comprehensive description of these phenomena going on in underdamped and overdamped pendula subject to additive and multiplicative periodic and random forces. No preliminary knowledge, such as complex mathematical or numerical methods, is required from a reader other than undergraduate courses in mathematical physics. A wide group of researchers, along with students and teachers will, thus, benefit from this definitive book on nonlinear dynamics.Table of ContentsPendulum Equations; Applications; Phase Space; Poincare Sections; Lyapunov Exponents; Spectral Analysis; Transition to Chaos; Analytic Methods (Feigenbaum, Melnikov); Parametric Driving; Brownian Motion; Overdamped; Spring; Double and Spherical Pendula.

Read more less

Book SynopsisThis is the first monograph dealing with the applications of the Lie group analysis to the modeling equations governing internal wave propagation in the deep ocean. A new approach to describe the nonlinear interactions of internal waves in the ocean is presented. While the central idea of the book is to investigate oceanic internal waves through the prism of Lie group analysis, it is also shown for the first time that internal wave beams, representing exact solutions to the equation of motion of stratified fluid, can be found by solving the given model as invariant solutions of nonlinear equations of motion. On the illustrative basis, it is also shown that the presence of the invariant solutions makes it possible to construct a more general class of disturbances, which represent wave beams propagating in certain direction coinciding with the beam energy.This book is designed for specialists in Geophysical Fluid Dynamics, Ocean and Atmospheric Modeling, as well as for researchers, teachers and students — mathematicians and nonmathematicians — interested in methods of applied group analysis for investigating nonlinear problems in physical, engineering and natural sciences. It can also serve as a textbook on practical applications of symmetries of nonlinear differential equations for graduate students in applied mathematics, physics and engineering.Table of ContentsInternal Waves in Stratified Fluid: Introduction; Governing Equations; Two Model Examples; Fundamentals of Lie Group Analysis: Calculus of Differential Algebra; Transformation Groups; Symmetries of Differential Equations; Applications of Symmetries; Group Analysis of Internal Waves: Generalities; Conservation Laws; Group Invariant Solutions.

Read more less

Book SynopsisThis collection of review articles is devoted to new developments in the study of chaotic dynamical systems with some open problems and challenges. The papers, written by many of the leading experts in the field, cover both the experimental and theoretical aspects of the subject. This edited volume presents a variety of fascinating topics of current interest and problems arising in the study of both discrete and continuous time chaotic dynamical systems. Exciting new techniques stemming from the area of nonlinear dynamical systems theory are currently being developed to meet these challenges. Presenting the state-of-the-art of the more advanced studies of chaotic dynamical systems, Frontiers in the Study of Chaotic Dynamical Systems with Open Problems is devoted to setting an agenda for future research in this exciting and challenging field.Table of ContentsProblems with Lorenz's Modeling and the Algorithm of Chaos Doctrine (S OuYang & Y Lin); Nonexistence of Chaotic Solutions of Nonlinear Differential Equations (L S Yao); Dynamics of Quadratic and Higher Degree Polynomial ODE Systems (J Heidel & F Zhang); On Chaotic and Hyperchaotic Complex Nonlinear Dynamical Systems (G M Mahmoud); On the Study of Chaotic Systems with Non-Horseshoe Template (A Ray et al.); Instability of Solutions of Fourth and Fifth Order Delay Differential Equations (C Tunc); Some Conjectures about the Synchronizability and the Topology of Networks (A Caneco et al.); Wavelet Study of Dynamical Systems using Partial Differential Equations (E B Postnikov); Combining the Dynamics of Discrete Dynamical Systems (J S Canovas); Code Structure for Pairs of Linear Maps with Some Open Problems (P Troshin); Recent Advances in Open Billiards with Some Open Problems (C P Dettmann); Open Problems in the Dynamics of the Expression of Gene Interaction Networks (L S Liebovitch & V Naudot); How to Transform a Type of Chaos in Dynamical Systems (E Zeraoulia & J C Sprott).

Read more less

Book SynopsisMemory is a universal function of organized matter. What is the mathematics of memory? How does memory affect the space-time behaviour of spatially extended systems? Does memory increase complexity? This book provides answers to these questions.It focuses on the study of spatially extended systems, i.e., cellular automata and other related discrete complex systems. Thus, arrays of locally connected finite state machines, or cells, update their states simultaneously, in discrete time, by the same transition rule. The classical dynamics in these systems is Markovian: only the actual configuration is taken into account to generate the next one.Generalizing the conventional view on spatially extended discrete dynamical systems evolution by allowing cells (or nodes) to be featured by some trait state computed as a function of its own previous state-values, the transition maps of the classical systems are kept unaltered, so that the effect of memory can be easily traced.The book demonstrates that discrete dynamical systems with memory are not only priceless tools for modeling natural phenomena but unique mathematical and aesthetic objects.Table of ContentsCellular Automata and Memory; Average Type Memory; Other Memories; Asynchrony and Probabilistic Rules; Cycles and Random Sequences; Three State Automata; Reversible Dynamics; Block Cellular Automata; Structurally Dynamic Systems; Boolean Networks; Coupled Layers; Continuous State Variable; Spatial Games.

Read more less

Book SynopsisThe work done in chaotic modeling and simulation during the last decades has changed our views of the world around us and has introduced new scientific tools, methods and techniques. Advanced topics of these achievements are included in this volume on Chaos Theory which focuses on Chaotic Modeling, Simulation and Applications of the nonlinear phenomena. This volume includes the best papers presented in the 3rd International Conference on CHAOS. This interdisciplinary conference attracted people from many scientific fields dealing with chaos, nonlinear dynamics, fractals and the works presented and the papers included here are of particular interest that could provide a broad understanding of chaos in its various forms.The chapters relate to many fields of chaos including Dynamical and Nonlinear Systems, Attractors and Fractals, Hydro-Fluid Dynamics and Mechanics, Chaos in Meteorology and Cosmology, Chaos in Biology and Genetics, Chaotic Control, Chaos in Economy and Markets, and Computer Composition and Chaotic Simulations, including related applications.

Read more less

Book SynopsisRobust chaos is defined by the absence of periodic windows and coexisting attractors in some neighborhoods in the parameter space of a dynamical system. This unique book explores the definition, sources, and roles of robust chaos. The book is written in a reasonably self-contained manner and aims to provide students and researchers with the necessary understanding of the subject. Most of the known results, experiments, and conjectures about chaos in general and about robust chaos in particular are collected here in a pedagogical form. Many examples of dynamical systems, ranging from purely mathematical to natural and social processes displaying robust chaos, are discussed in detail. At the end of each chapter is a set of exercises and open problems (more than 260 in the whole book) intended to reinforce the ideas and provide additional experiences for both readers and researchers in nonlinear science in general, and chaos theory in particular.Table of ContentsPoincare Map Technique, Smale Horseshoe and Symbolic Dynamics; Robustness of Chaos; Statistical Properties of Chaotic Attractors; Structural Stability; Transversality, Invariant Foliation, and the Shadowing Lemma; Chaotic Attractors with Hyperbolic Structure; Robust Chaos in Hyperbolic Systems; Lorenz-Type Systems; Robust Chaos in the Lorenz-Type Systems; No Robust Chaos in Quasi-Attractors; Robust Chaos in One-Dimensional Maps; Robust Chaos in 2-D Piecewise Smooth Maps.

Read more less

Book SynopsisThe book Complexity and Control: Towards a Rigorous Behavioral Theory of Complex Dynamical Systems is a graduate-level monographic textbook, intended to be a novel and rigorous contribution to modern Complexity Theory.This book contains 11 chapters and is designed as a one-semester course for engineers, applied and pure mathematicians, theoretical and experimental physicists, computer and economic scientists, theoretical chemists and biologists, as well as all mathematically educated scientists and students, both in industry and academia, interested in predicting and controlling complex dynamical systems of arbitrary nature.

Read more less

Book SynopsisThis book aims to provide a lively working knowledge of the thermodynamic control of microscopic simulations, while summarizing the historical development of the subject, along with some personal reminiscences. Many computational examples are described so that they are well-suited to learning by doing. The contents enhance the current understanding of the reversibility paradox and are accessible to advanced undergraduates and researchers in physics, computation, and irreversible thermodynamics.Table of ContentsOverview of Atomistic Mechanics; Formulating Atomistic Simulations; Thermodynamics, Statistical Mechanics, and Temperature; Continuum Mechanics: Continuity, Stress, Heat Flux, Applications; Numerical Molecular Dynamics and Chaos; Time-Reversible Atomistic Thermostats; Key Results from Nonequilibrium Simulations; Second Law, Reversibility, Instability; Outlook for Progress and Life on Earth;

Read more less

Book SynopsisThis book focuses on the computational analysis of nonlinear vibrations of structural members (beams, plates, panels, shells), where the studied dynamical problems can be reduced to the consideration of one spatial variable and time. The reduction is carried out based on a formal mathematical approach aimed at reducing the problems with infinite dimension to finite ones. The process also includes a transition from governing nonlinear partial differential equations to a set of finite number of ordinary differential equations.Beginning with an overview of the recent results devoted to the analysis and control of nonlinear dynamics of structural members, placing emphasis on stability, buckling, bifurcation and deterministic chaos, simple chaotic systems are briefly discussed. Next, bifurcation and chaotic dynamics of the Euler-Bernoulli and Timoshenko beams including the geometric and physical nonlinearity as well as the elastic-plastic deformations are illustrated. Despite the employed classical numerical analysis of nonlinear phenomena, the various wavelet transforms and the four Lyapunov exponents are used to detect, monitor and possibly control chaos, hyper-chaos, hyper-hyper-chaos and deep chaos exhibited by rectangular plate-strips and cylindrical panels.The book is intended for post-graduate and doctoral students, applied mathematicians, physicists, teachers and lecturers of universities and companies dealing with a nonlinear dynamical system, as well as theoretically inclined engineers of mechanical and civil engineering.Table of ContentsBifurcation and Chaos of Structural Members; Introduction to Chaos and Wavelets; Quantifying Chaos; Simple Chaotic Models; Dissipative Systems; Euler-Bernoulli Beams; Planar Beams; Beams and Temperature/Electric Fields; Beams with Elasto-Plastic Deformations; Multi-Layer Beams; Timoshenko and Sheremetev-Pelekh Beams; Panels and Cylindrical Panels; Plates and Shells; Flexible Axially-Symmetric Shells;

Read more less

Book SynopsisThis book provides a good coverage of the recent developments and future directions in the study of dissipative systems. The primary thrust here is in exposing the reader to the frontiers of chaos, pointing out clues for further work in nonlinear science. With the aid of various types of mappings, the collapse of tori is investigated. The book contains much valuable introductory material and copious reference lists. Some notes on the historical development of the subject are interspersed in this volume.Trade Review"It contains a wealth of numerical observations which reveal of lot of different phenomena. It is not written for the uninitiated but I recommend it to anyone who thinks that the world is simple. Your illusions will be shattered." R S Mackay Journal of Modern Optics (USA), 1987

Read more less