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A discrete dynamical system can be characterized as an iterated function. Given the efficiency with which computers can do iteration, it is now possible for anyone with access to a personal computer to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. The level of presentation is suitable for advanced undergraduates who have completed a year of college-level calculus. Concepts from calculus are reviewed as necessary. Mathematica programs that illustrate the dynamics and that will aid the student in doing the exercises are included in the appendix. In this second edition, the covered topics are rearranged to make the text more flexible. In particular, the material on symbolic dynamics is now optional and the book can easily be used for a semester course dealing exclusively with functions of a real variable. Alternatively, the basic properties of dynamical systems can be introduced using functions of a real variable and then the reader can skip directly to the material on the dynamics of complex functions. Additional changes include the simplification of several proofs; a thorough review and expansion of the exercises; and substantial improvement in the efficiency of the Mathematica programs. ... Read more

Customer Reviews (2)

Amazing motivational introduction
An excellent short motivational reading / introductory course material. Starts ground up and uses very simple hands-on examples to demonstrate the fascinating behavior of non-linear dynamic systems (chaos, fractals, etc). Covers most of the basic methodology.
The book also has a lot of excellent exercises for reinforcing the material, for different difficulty levels.

An excellent book
This is an excellent book for teaching for mathematics students. Besides the standard material it contains a chapter about some numerical methods as dynamical systems. I strongly propose that a third updated edition is made. ... Read more

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The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics.This volume collects the Lectures held at the 2008 CIME session on "Holomorphic Dynamical Systems" held in Cetraro, Italy. This CIME Course focused on a number of important topics in the study of discrete andcontinuous dynamical systems, including both local and global aspects, providing a fascinating introduction to many key problems in current research. The contributions provide an ample description of the phenomena occurring in central themes of holomorphic dynamics such as automorphisms and meromorphic self-maps of projective spaces, of entire maps on complex spaces and holomorphic foliations in surfaces and higher dimensional manifolds, elaborating on the different techniques used and familiarizing readers with the latest findings on current research topics. ... Read more

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In this book the theory of hyperbolic sets is developed, bothfor diffeomorphisms and flows, with an emphasis on shadowing. We showthat hyperbolic sets are expansive and have the shadowing property.Then we use shadowing to prove that hyperbolic sets are robust underperturbation, that they have an asymptotic phase property and alsothat the dynamics near a transversal homoclinic orbit is chaotic.
It turns out that chaotic dynamical systems arising in practice arenot quite hyperbolic. However, they possess enough hyperbolicity toenable us to use shadowing ideas to give computer-assisted proofs thatcomputed orbits of such systems can be shadowed by true orbits forlong periods of time, that they possess periodic orbits of longperiods and that it is really true that they are chaotic.
Audience: This book is intended primarily for research workersin dynamical systems but could also be used in an advanced graduatecourse taken by students familiar with calculus in Banach spaces andwith the basic existence theory for ordinary differential equations. ... Read more

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Based on the results of over 10 years of research and development by the authors, this book presents a broad cross section of dynamic programming (DP) techniques applied to the optimization of dynamical systems. The main goal of the research effort was to develop a robust path planning/trajectory optimization tool that did not require an initial guess. The goal was partially met with a combination of DP and homotopy algorithms. DP algorithms are presented here with a theoretical development, and their successful application to variety of practical engineering problems is emphasized.Applied Dynamic Programming for Optimization of Dynamical Systems presents applications of DP algorithms that are easily adapted to the reader’s own interests and problems. The book is organized in such a way that it is possible for readers to use DP algorithms before thoroughly comprehending the full theoretical development. A general architecture is introduced for DP algorithms emphasizing the solution to nonlinear problems. DP algorithm development is introduced gradually with illustrative examples that surround linear systems applications. Many examples and explicit design steps applied to case studies illustrate the ideas and principles behind DP algorithms.DP algorithms potentially address a wide class of applications composed of many different physical systems described by dynamical equations of motion that require optimized trajectories for effective maneuverability. The DP algorithms determine control inputs and corresponding state histories of dynamic systems for a specified time while minimizing a performance index. Constraints may be applied to the final states of the dynamic system or to the states and control inputs during the transient portion of the maneuver. List of Figures; Preface; List of Tables; Chapter 1: Introduction; Chapter 2: Constrained Optimization; Chapter 3: Introduction to Dynamic Programming; Chapter 4: Advanced Dynamic Programming; Chapter 5: Applied Case Studies; Appendix A: Mathematical Supplement; Appendix B: Applied Case Studies - MATLAB Software Addendum; Bibliography; Index.Physicists and mechanical, electrical, aerospace, and industrial engineers will find this book enormously useful.It will also appeal to research scientists and engineering students who have a background in dynamics and control and are able to develop and apply the DP algorithms to their particular problems. This book is suitable as a reference or supplemental textbook for graduate courses in optimization of dynamical and control systems. ... Read more

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Sequential Dynamical Systems (SDS) are a class of discrete dynamical systems which significantly generalize many aspects of systems such as cellular automata, and provide a framework for studying dynamical processes over graphs.

This text is the first to provide a comprehensive introduction to SDS. Driven by numerous examples and thought-provoking problems, the presentation offers good foundational material on finite discrete dynamical systems which leads systematically to an introduction of SDS. Techniques from combinatorics, algebra and graph theory are used to study a broad range of topics, including reversibility, the structure of fixed points and periodic orbits, equivalence, morphisms and reduction. Unlike other books that concentrate on determining the structure of various networks, this book investigates the dynamics over these networks by focusing on how the underlying graph structure influences the properties of the associated dynamical system.

This book is aimed at graduate students and researchers in discrete mathematics, dynamical systems theory, theoretical computer science, and systems engineering who are interested in analysis and modeling of network dynamics as well as their computer simulations. Prerequisites include knowledge of calculus and basic discrete mathematics. Some computer experience and familiarity with elementary differential equations and dynamical systems are helpful but not necessary.

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His research in dynamics constitutes the middle period of Birkhoff's scientific career, that of maturity and greatest power. --Yearbook of the American Philosophical Society The author's great book ... is well known to all, and the diverse active modern developments in mathematics which have been inspired by this volume bear the most eloquent testimony to its quality and influence. --Zentralblatt MATH In 1927, G. D. Birkhoff wrote a remarkable treatise on the theory of dynamical systems that would inspire many later mathematicians to do great work. To a large extent, Birkhoff was writing about his own work on the subject, which was itself strongly influenced by Poincare's approach to dynamical systems. With this book, Birkhoff also demonstrated that the subject was a beautiful theory, much more than a compendium of individual results. The influence of this work can be found in many fields, including differential equations, mathematical physics, and even what is now known as Morse theory. The present volume is the revised 1966 reprinting of the book, including a new addendum, some footnotes, references added by Jurgen Moser, and a special preface by Marston Morse. Although dynamical systems has thrived in the decades since Birkhoff's book was published, this treatise continues to offer insight and inspiration for still more generations of mathematicians. ... Read more

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This work represents the current state of the most important economic applications of dynamical systems theory. The volume features not only a survey of modern analytical approaches and nonlinear models but also an anthology of the numerical and graphical methods (represented in 90 illustrations) that are essential to an understanding of dynamical behavior. The first chapters present accessible (though not elementary) introductions to local analysis and the linear approximation, deep issues regarding chaos and complexity, the ergodic approach and probabilistic properties of dynamical systems. The following chapters provide a unique collection of current nonlinear models in micro and macroeconomics, including those based on overlapping generations, heterogeneous agents, evolutionary financial markets, oligopolist and duopolist games.

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This book is based on a graduate course for scientists and engineers. ... Read more

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This book is addressed to graduate students and researchers in mathematics who are interested in mathematical and theoretical physics, symplectic geometry, mechanics, and (geometric) quantization.

The aim of the book is to treat all three basic theories of physics, namely, classical mechanics, statistical mechanics, and quantum mechanics from the same perspective, that of symplectic geometry, thus showing the unifying power of the symplectic geometric approach. Reading this book will give the reader a deep understanding of the interrelationships between the three basic theories of physics.

The first two chapters provide the necessary mathematical background in differential geometry, Lie groups, and symplectic geometry. In Chapter 3 a coherent symplectic description of Galilean and relativistic mechanics is given, culminating in the classification of elementary particles (relativistic and non-relativistic, with or without spin, with or without mass). In Chapter 4 statistical mechanics is put into symplectic form, finishing with a symplectic description of the kinetic theory of gases and the computation of specific heats. Finally in Chapter 5 the author presents his theory of geometric quantization. Highlights of this chapter are the derivation of the various wave equations and the construction of the Fock space. ... Read more

Customer Reviews (1)

Breathtaking Synthesis of Physics
The influence of this book is already large, and with this new edition of the English translation, destined to get bigger. Schlomo Sternberg at Harvard has already re-invented the way Undergrad Physics is usually taught with his Symplectic Approach, due in no small measure to his reading of this very book in its previous (French) incarnation. But Souriau is not only readable, and 'followable' in his expositions, the results are so delightful. The only complaint I would make is with the book editors' use of "cutesy" icons (spades, hearts, clubs, diamonds, something that looks like a bugle - for BIG results, one imagines hearing a fanfare every time one sees it), these sometimes get to be distracting. However, the editors' notes are useful and worthwhile. I fully believe the 'program' first enunciated (as far as I'm aware) in this book in 1968, will become more and more a dominant paradigm for physics in the 21st century. ... Read more

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The conjoining of nonlinear dynamics and biology has brought about significant advances in both areas, with nonlinear dynamics providing a tool for understanding biological phenomena and biology stimulating developments in the theory of dynamical systems. This research monograph provides an introduction to the theory of nonautonomous semiflows with applications to population dynamics. It develops dynamical system approaches to various evolutionary equations such as difference, ordinary, functional, and partial differential equations, and pays more attention to periodic and almost periodic phenomena. The presentation includes persistence theory, monotone dynamics, periodic and almost periodic semiflows, traveling waves, and global analysis of typical models in population biology. Research mathematicians working with nonlinear dynamics, particularly those interested in applications to biology, will find this book useful. It may also be used as a textbook or as supplementary reading for a graduate special topics course on the theory and applications of dynamical systems.

Dr. Xiao-Qiang Zhao is a professor in applied mathematics at Memorial University of Newfoundland, Canada. His main research interests involve applied dynamical systems, nonlinear differential equations, and mathematical biology. He is the author of more than 40 papers and his research has played an important role in the development of the theory of periodic and almost periodic semiflows and their applications.

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Topology, the foundation of modern analysis, arose historically as a way to organize ideas like compactness and connectedness which had emerged from analysis. Similarly, recent work in dynamical systems theory has both highlighted certain topics in the pre-existing subject of topological dynamics (such as the construction of Lyapunov functions and various notions of stability) and also generated new concepts and results (such as attractors, chain recurrence, and basic sets). This book collects these results, both old and new, and organizes them into a natural foundation for all aspects of dynamical systems theory. No existing book is comparable in content or scope. Requiring background in point-set topology and some degree of ``mathematical sophistication'', Akin's book serves as an excellent textbook for a graduate course in dynamical systems theory. In addition, Akin's reorganization of previously scattered results makes this book of interest to mathematicians and other researchers who use dynamical systems in their work. ... Read more

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Breadth of scope is unique

Author is a widely-known and successful textbook author

Unlike many recent textbooks on chaotic systems that have superficial treatment, this book provides explanations of the deep underlying mathematical ideas

No technical proofs, but an introduction to the whole field that is based on the specific analysis of carefully selected examples

Includes a section on cellular automata

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This book unites the study of dynamical systems and numerical solution of differential equations. The first three chapters contain the elements of the theory of dynamical systems and the numerical solution of initial-value problems.In the remaining chapters, numerical methods are formulted as dynamical systems and the convergence and stability properties of the methods are examined.Topics studied include the stability of numerical methods for contractive, dissipative, gradient and Hamiltonian systems together with the convergence properties of equilibria, periodic solutions and strage attractors under numerical approximation.This book will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems. ... Read more

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Largely self-contained, this is an introduction to the mathematical structures underlying models of systems whose state changes with time, and which therefore may exhibit "chaotic behavior." The first portion of the book is based on lectures given at the University of London and covers the background to dynamical systems, the fundamental properties of such systems, the local bifurcation theory of flows and diffeomorphisms and the logistic map and area-preserving planar maps.The authors then go on to consider current research in this field such as the perturbation of area-preserving maps of the plane and the cylinder.The text contains many worked examples and exercises, many with hints.It will be a valuable first textbook for senior undergraduate and postgraduate students of mathematics, physics, and engineering. ... Read more

Customer Reviews (2)

Excellent Introduction to the Subject
covers most important areas of the subject with a clear yet rigorous approach.Advanced text better suited for graduate students in applied math.It promises as a must for anyone serious about the subject

Excellent focus on what is important
Dynamical systems is a vast subject to which no single book can provide an adequate introduction, but the authors do an excellent job of focusing on what is important and avoiding the temptation to go off on enticing tangents. Their treatment is clear, and this book is highly recommended for any student seeking a solid foundation for further work. ... Read more

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Dynamical Systems with Applications Using Mathematica^® provides an introduction to the theory of dynamical systems with the aid of the Mathematica computer algebra package. The book has a very hands-on approach and takes the reader from basic theory to recently published research material. Emphasized throughout are numerous applications to biology, chemical kinetics, economics, electronics, epidemiology, nonlinear optics, mechanics, population dynamics, and neural networks.

Throughout the book, the author has focused on breadth of coverage rather than fine detail, with theorems and proofs being kept to a minimum. The first part of the book deals with continuous systems using ordinary differential equations, while the second part is devoted to the study of discrete dynamical systems. Exercises are included at the end of every chapter. Both textbooks and research papers are presented in the list of references.

Working Mathematica notebooks will be available at

http://library.wolfram.com/infocenter/Books/AppliedMathematics/.

The book is intended for senior undergraduate and graduate students as well as working scientists in applied mathematics, the natural sciences, and engineering. The material is also accessible to readers with a general mathematical background. Many chapters of the book are especially useful as reference material for senior undergraduate independent project work.

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This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey "Principal Structures" of Volume 1A.

The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations).

. Written by experts in the field.
. The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources.
. The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems. ... Read more

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From the reviews of the first edition:"... Here ... a wealth of material is displayed for us, too much to even indicate in a review. ... Your reviewer was very impressed by the contents of both volumes (EMS 2 and 4), recommending them without any restriction." Mededelingen van het Wiskundig genootshap 1992

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Nonautonomous dynamical systems provide a mathematical framework for temporally changing phenomena, where the law of evolution varies in time due to seasonal, modulation, controlling or even random effects. Our goal is to provide an approach to the corresponding geometric theory of nonautonomous discrete dynamical systems in infinite-dimensional spaces by virtue of 2-parameter semigroups (processes). These dynamical systems are generated by implicit difference equations, which explicitly depend on time. Compactness and dissipativity conditions are provided for such problems in order to have attractors using the natural concept of pullback convergence. Concerning a necessary linear theory, our hyperbolicity concept is based on exponential dichotomies and splittings. This concept is in turn used to construct nonautonomous invariant manifolds, so-called fiber bundles, and deduce linearization theorems. The results are illustrated using temporal and full discretizations of evolutionary differential equations. ... Read more

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Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational, accuracy, and storage capabilities, model reduction (system approximation) is often necessary. Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications.This book is for anyone interested in model reduction. Graduate students and researchers in the fields of system and control theory, numerical analysis, and the theory of partial differential equations/computational fluid dynamics will find it an excellent reference.ContentsList of Figures; Foreword; Preface; How to Use this Book; Part I: Introduction. Chapter 1: Introduction; Chapter 2: Motivating Examples; Part II: Preliminaries. Chapter 3: Tools from Matrix Theory; Chapter 4: Linear Dynamical Systems: Part 1; Chapter 5: Linear Dynamical Systems: Part 2; Chapter 6: Sylvester and Lyapunov equations; Part III: SVD-based Approximation Methods. Chapter 7: Balancing and balanced approximations; Chapter 8: Hankel-norm Approximation; Chapter 9: Special topics in SVD-based approximation methods; Part IV: Krylov-based Approximation Methods; Chapter 10: Eigenvalue Computations; Chapter 11: Model Reduction Using Krylov Methods; Part V: SVD–Krylov Methods and Case Studies. Chapter 12: SVD–Krylov Methods; Chapter 13: Case Studies; Chapter 14: Epilogue; Chapter 15: Problems; Bibliography; Index. ... Read more

Customer Reviews (1)

An Excellent Book on Model Order Reduction
Dr. Athanasios C. Antoulas is one of the world's leading authorities on model order reduction. In this book he clearly differentiated Singular Value Decomposition (SVD) methods and Krylov subspace projection methods and their explanation in part III and IV is excellent. The fist two chapter of introduction and preliminaries give you sufficient background to understand Part III and IV. The book does require some familiarity with numerical linear algebra and may be difficult for a new comer in the area to follow, however the way it is written it can be used both by engineers and scientist. A must-have book for any one working on model order reduction or dealing with large scale dynamical system. By reading this book I clearly understood the concept of reachability and observability and how it is applied to better understand a dynamical system. Overall a book with excellent source of knowledge, both as text and reference book. My recommendation definitely 5 stars. ... Read more

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

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