A First Course in Discrete Dynamical Systems

Author: Richard A. Holmgren

Publisher: Springer Science & Business Media

ISBN: 1441987320

Category: Mathematics

Page: 223

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Given the ease with which computers can do iteration it is now possible for almost anyone 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. Mathematica programs that illustrate the dynamics are included in an appendix.

Discrete Dynamical Systems, Bifurcations and Chaos in Economics

Author: Wei-Bin Zhang

Publisher: Elsevier

ISBN: 9780080462462

Category: Mathematics

Page: 460

View: 7180

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This book is a unique blend of difference equations theory and its exciting applications to economics. It deals with not only theory of linear (and linearized) difference equations, but also nonlinear dynamical systems which have been widely applied to economic analysis in recent years. It studies most important concepts and theorems in difference equations theory in a way that can be understood by anyone who has basic knowledge of calculus and linear algebra. It contains well-known applications and many recent developments in different fields of economics. The book also simulates many models to illustrate paths of economic dynamics. A unique book concentrated on theory of discrete dynamical systems and its traditional as well as advanced applications to economics Mathematical definitions and theorems are introduced in a systematic and easily accessible way Examples are from almost all fields of economics; technically proceeding from basic to advanced topics Lively illustrations with numerous figures Numerous simulation to see paths of economic dynamics Comprehensive treatment of the subject with a comprehensive and easily accessible approach

Discrete Dynamical Systems

Author: Oded Galor

Publisher: Springer Science & Business Media

ISBN: 3540367764

Category: Business & Economics

Page: 153

View: 7991

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

Discovering Discrete Dynamical Systems

Author: Aimee S. A. Johnson,Kathleen M. Madden,Ayşe A. Şahin

Publisher: The Mathematical Association of America

ISBN: 0883857936

Category: Mathematics

Page: 130

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A dynamical system is a collection of possible states and a rule (or rules) that describes evolution of these states over time. The main purpose of this book is to introduce important concepts in dynamical systems - including fixed and periodic points, attractors and repellers, chaos and fractals - in a way that encourages readers to explore, discover , and learn independently. The book differs from other dynamical system textbooks in that topics have been carefully chosen both to give a coherent introduction to dynamical systems and to support inquiry-based learning.

Discrete Dynamical Systems

Theory and Applications

Author: James T. Sandefur

Publisher: Oxford University Press

ISBN: 9780198533849

Category: Chaotic behavior in systems.

Page: 445

View: 9350

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Discrete dynamics is the study of change. In particular, it shows how to translate real world situations into the language of mathematics. With the increase in computational ability and the recent interest in chaos, discrete dynamics has emerged as an important area of mathematical study. This text is the first to provide an elementary introduction to the world of dynamical systems. The aim of the text is to explain both the wide variety of techniques used to study dynamical systems and their many applications in areas ranging from population growth to problems in genetics. This investigation leads to the fruitful concepts of stability, strange attractors, chaos, and fractals. Very little previous mathematical knowledge is assumed and students with an elementary exposure to calculus and linear algebra will be able to follow the text easily. A large number of worked examples and exercises are provided to assist instruction. Throughout, students are encouraged to experiment with models of dynamical systems on computers and explore this fascinating area of mathematics on their own.

Introduction to Discrete Dynamical Systems and Chaos

Author: Mario Martelli

Publisher: John Wiley & Sons

ISBN: 1118031121

Category: Mathematics

Page: 344

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A timely, accessible introduction to the mathematics of chaos. The past three decades have seen dramatic developments in the theory of dynamical systems, particularly regarding the exploration of chaotic behavior. Complex patterns of even simple processes arising in biology, chemistry, physics, engineering, economics, and a host of other disciplines have been investigated, explained, and utilized. Introduction to Discrete Dynamical Systems and Chaos makes these exciting and important ideas accessible to students and scientists by assuming, as a background, only the standard undergraduate training in calculus and linear algebra. Chaos is introduced at the outset and is then incorporated as an integral part of the theory of discrete dynamical systems in one or more dimensions. Both phase space and parameter space analysis are developed with ample exercises, more than 100 figures, and important practical examples such as the dynamics of atmospheric changes and neural networks. An appendix provides readers with clear guidelines on how to use Mathematica to explore discrete dynamical systems numerically. Selected programs can also be downloaded from a Wiley ftp site (address in preface). Another appendix lists possible projects that can be assigned for classroom investigation. Based on the author's 1993 book, but boasting at least 60% new, revised, and updated material, the present Introduction to Discrete Dynamical Systems and Chaos is a unique and extremely useful resource for all scientists interested in this active and intensely studied field. An Instructor's Manual presenting detailed solutions to all the problems in the book is available upon request from the Wiley editorial department.

Geometric Methods for Discrete Dynamical Systems

Author: Robert W. Easton

Publisher: Oxford University Press

ISBN: 9780195359046

Category: Science

Page: 176

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This book looks at dynamics as an iteration process where the output of a function is fed back as an input to determine the evolution of an initial state over time. The theory examines errors which arise from round-off in numerical simulations, from the inexactness of mathematical models used to describe physical processes, and from the effects of external controls. The author provides an introduction accessible to beginning graduate students and emphasizing geometric aspects of the theory. Conley's ideas about rough orbits and chain-recurrence play a central role in the treatment. The book will be a useful reference for mathematicians, scientists, and engineers studying this field, and an ideal text for graduate courses in dynamical systems.

Chaos in Discrete Dynamical Systems

A Visual Introduction in 2 Dimensions

Author: Ralph Abraham,Laura Gardini,Christian Mira

Publisher: Springer Science & Business Media

ISBN: 1461219361

Category: Mathematics

Page: 246

View: 6765

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The materials in the book and on the accompanying disc are not solely developed with only the researcher and professional in mind, but also with consideration for the student: most of this material has been class-tested by the authors. The book is packed with some 100 computer graphics to illustrate the material, and the CD-ROM contains full-colour animations tied directly to the subject matter of the book itself. The cross-platform CD also contains the program ENDO, which enables users to create their own 2-D imagery with X-Windows. Maple scripts are provided to allow readers to work directly with the code from which the graphics in the book were taken.

An Introduction to Dynamical Systems

Continuous and Discrete

Author: Rex Clark Robinson

Publisher: American Mathematical Soc.

ISBN: 0821891359

Category: Mathematics

Page: 733

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This book gives a mathematical treatment of the introduction to qualitative differential equations and discrete dynamical systems. The treatment includes theoretical proofs, methods of calculation, and applications. The two parts of the book, continuous time of differential equations and discrete time of dynamical systems, can be covered independently in one semester each or combined together into a year long course. The material on differential equations introduces the qualitative or geometric approach through a treatment of linear systems in any dimension. There follows chapters where equilibria are the most important feature, where scalar (energy) functions is the principal tool, where periodic orbits appear, and finally, chaotic systems of differential equations. The many different approaches are systematically introduced through examples and theorems. The material on discrete dynamical systems starts with maps of one variable and proceeds to systems in higher dimensions. The treatment starts with examples where the periodic points can be found explicitly and then introduces symbolic dynamics to analyze where they can be shown to exist but not given in explicit form. Chaotic systems are presented both mathematically and more computationally using Lyapunov exponents. With the one-dimensional maps as models, the multidimensional maps cover the same material in higher dimensions. This higher dimensional material is less computational and more conceptual and theoretical. The final chapter on fractals introduces various dimensions which is another computational tool for measuring the complexity of a system. It also treats iterated function systems which give examples of complicated sets. In the second edition of the book, much of the material has been rewritten to clarify the presentation. Also, some new material has been included in both parts of the book. This book can be used as a textbook for an advanced undergraduate course on ordinary differential equations and/or dynamical systems. Prerequisites are standard courses in calculus (single variable and multivariable), linear algebra, and introductory differential equations.

Analysis and Modelling of Discrete Dynamical Systems

Author: Daniel Benest,Claude Froeschle

Publisher: CRC Press

ISBN: 9789056996253

Category: Computers

Page: 344

View: 685

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The theory of dynamical systems, or mappings, plays an important role in various disciplines of modern physics, including celestial mechanics and fluid mechanics. This comprehensive introduction to the general study of mappings has particular emphasis on their applications to the dynamics of the solar system. The book forms a bridge between continuous systems, which are suited to analytical developments and to discrete systems, which are suitable for numerical exploration. Featuring chapters based on lectures delivered at the School on Discrete Dynamical Systems (Aussois, France, February 1996) the book contains three parts - Numerical Tools and Modelling, Analytical Methods, and Examples of Application. It provides a single source of information that, until now, has been available only in widely dispersed journal articles.

Advances in Discrete Dynamical Systems

Author: Saber Elaydi

Publisher: Advanced Studies in Pure Mathe

ISBN: N.A

Category: Mathematics

Page: 398

View: 7604

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This volume contains the proceedings of talks presented at the 11th International Conference on Difference Equations and Applications (ICDEA 2006). ICDEA 2006 was held on July 2006 in Kyoto at the 15th MSJ International Research Institute. These proceedings comprise new results at the leading edge of many areas in difference equations and discrete dynamical systems and their various applications to the sciences, engineering, physics, and economics.

Discrete dynamical systems and chaos

Author: Mario Martelli

Publisher: Chapman & Hall/CRC

ISBN: N.A

Category: Mathematics

Page: 282

View: 7438

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In a balanced presentation, this Monograph presents definitions and results on dynamical systems and chaos in a manner accessible to undergraduates in a range of disciplines with of theoretical results and practical applications.

Discrete Dynamical Systems and Chaotic Machines

Theory and Applications

Author: Jacques M. Bahi,Christophe Guyeux

Publisher: CRC Press

ISBN: 1466554509

Category: Computers

Page: 230

View: 2705

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For computer scientists, especially those in the security field, the use of chaos has been limited to the computation of a small collection of famous but unsuitable maps that offer no explanation of why chaos is relevant in the considered contexts. Discrete Dynamical Systems and Chaotic Machines: Theory and Applications shows how to make finite machines, such as computers, neural networks, and wireless sensor networks, work chaotically as defined in a rigorous mathematical framework. Taking into account that these machines must interact in the real world, the authors share their research results on the behaviors of discrete dynamical systems and their use in computer science. Covering both theoretical and practical aspects, the book presents: Key mathematical and physical ideas in chaos theory Computer science fundamentals, clearly establishing that chaos properties can be satisfied by finite state machines Concrete applications of chaotic machines in computer security, including pseudorandom number generators, hash functions, digital watermarking, and steganography Concrete applications of chaotic machines in wireless sensor networks, including secure data aggregation and video surveillance Until the authors’ recent research, the practical implementation of the mathematical theory of chaos on finite machines raised several issues. This self-contained book illustrates how chaos theory enables the study of computer security problems, such as steganalysis, that otherwise could not be tackled. It also explains how the theory reinforces existing cryptographically secure tools and schemes.

Discrete Dynamical Systems and Difference Equations with Mathematica

Author: Mustafa R.S. Kulenovic,Orlando Merino

Publisher: CRC Press

ISBN: 1420035355

Category: Mathematics

Page: 360

View: 6120

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Following the work of Yorke and Li in 1975, the theory of discrete dynamical systems and difference equations developed rapidly. The applications of difference equations also grew rapidly, especially with the introduction of graphical-interface software that can plot trajectories, calculate Lyapunov exponents, plot bifurcation diagrams, and find basins of attraction. Modern computer algebra systems have opened the door to the use of symbolic calculation for studying difference equations. This book offers an introduction to discrete dynamical systems and difference equations and presents the Dynamica software. Developed by the authors and based on Mathematica, Dynamica provides an easy-to-use collection of algebraic, numerical, and graphical tools and techniques that allow users to quickly gain the ability to: Find and classify the stability character of equilibrium and periodic points Perform semicycle analysis of solutions Calculate and visualize invariants Calculate and visualize Lyapunov functions and numbers Plot bifurcation diagrams Visualize stable and unstable manifolds Calculate Box Dimension While it presents the essential theoretical concepts and results, the book's emphasis is on using the software. The authors present two sets of Dynamica sessions: one that serves as a tutorial of the different techniques, the other features case studies of well-known difference equations. Dynamica and notebooks corresponding to particular chapters are available for download from the Internet.

Gewöhnliche Differentialgleichungen

Author: Vladimir I. Arnold

Publisher: Springer-Verlag

ISBN: 3642564801

Category: Mathematics

Page: 344

View: 3157

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nen (die fast unverändert in moderne Lehrbücher der Analysis übernommen wurde) ermöglichten ihm nach seinen eigenen Worten, "in einer halben Vier telstunde" die Flächen beliebiger Figuren zu vergleichen. Newton zeigte, daß die Koeffizienten seiner Reihen proportional zu den sukzessiven Ableitungen der Funktion sind, doch ging er darauf nicht weiter ein, da er zu Recht meinte, daß die Rechnungen in der Analysis bequemer auszuführen sind, wenn man nicht mit höheren Ableitungen arbeitet, sondern die ersten Glieder der Reihenentwicklung ausrechnet. Für Newton diente der Zusammenhang zwischen den Koeffizienten der Reihe und den Ableitungen eher dazu, die Ableitungen zu berechnen als die Reihe aufzustellen. Eine von Newtons wichtigsten Leistungen war seine Theorie des Sonnensy stems, die in den "Mathematischen Prinzipien der Naturlehre" ("Principia") ohne Verwendung der mathematischen Analysis dargestellt ist. Allgemein wird angenommen, daß Newton das allgemeine Gravitationsgesetz mit Hilfe seiner Analysis entdeckt habe. Tatsächlich hat Newton (1680) lediglich be wiesen, daß die Bahnkurven in einem Anziehungsfeld Ellipsen sind, wenn die Anziehungskraft invers proportional zum Abstandsquadrat ist: Auf das Ge setz selbst wurde Newton von Hooke (1635-1703) hingewiesen (vgl. § 8) und es scheint, daß es noch von weiteren Forschern vermutet wurde.

Difference Equations and Discrete Dynamical Systems

Proceedings of the 9th International Conference, University of Southern California, Los Angeles, California, USA, 2-7 August 2004

Author: Linda J. S. Allen,Bernd Aulbach,Saber Elaydi

Publisher: World Scientific Publishing Company Incorporated

ISBN: 9789812565204

Category: Mathematics

Page: 324

View: 8873

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A discrete-time Beverton-Holt competition model / Azmy S. Ackleh, Youssef M. Dib and Sophia R.-J. Jang -- A dynamic analysis of the Bush fiscal policy / Richard H. Day and Chengyu Yang -- A hybrid approximation to certain delay differential equation with a constant delay / George Seifert -- Compulsory asymptotic behavior of solutions of two-dimensional systems of difference equations / Josef Diblík and Irena R°uz̮ic̮ková -- Discrete models of differential equations : the roles of dynamic consistency and positivity / Ronald E. Mickens -- Enveloping implies global stability / Paul Cull -- Global asymptotic stability in the Jia Li model for genetically altered mosquitoes / Robert J. Sacker and Hubertus F. von Bremen -- Global behavior of solutions of a nonlinear second-order nonautonomous difference equation / Vlajko L. Kocic -- How can three species coexist in a periodic chemostat? Mathematical and numerical study / Shinji Nakaoka and Yasuhiro Takeuchi -- Information-theoretic measures of discrete orthogonal polynomials / Jesus Sanchez Dehesa ... [et al.] -- Local approximation of invariant fiber bundles : an algorithmic approach / Christian Pötzsche and Martin Rasmussen -- Necessary and sufficient conditions for oscillation of coupled nonlinear discrete systems / Serena Matucci and Pavel R̮ehák -- Non-standard finite difference methods for dissipative singular perturbation problems / Jean M.-S. Lubuma and Kailash C. Patidar -- On a class of generalized autoregressive processes / Kamal C. Chanda -- On [symbol] with period-two coefficients / Carol H. Gibbons and Carol B. Overdeep -- Periodically forced nonlinear difference equations with delay / Abdul-Aziz Yakubu -- Regularity of difference equations / Jarmo Hietarinta -- Robustness in difference equations / Jack K. Hale -- Solvability of the discrete LQR-problem under minimal assumptions / Roman Hilscher and Vera Zeidan -- Some discrete competition models and the principle of competitive exclusion / Jim M. Cushing and Sheree Le Varge -- Stability under constantly acting perturbations for difference equations and averaging / Vladimir Burd -- Symbolic dynamics in the study of bursting electrical activity / Jorge Duarte, Jose Sousa Ramos and Luis Silva

Theory and Applications of Difference Equations and Discrete Dynamical Systems

ICDEA, Muscat, Oman, May 26 - 30, 2013

Author: Ziyad AlSharawi,Jim M. Cushing,Saber Elaydi

Publisher: Springer

ISBN: 3662441403

Category: Mathematics

Page: 222

View: 1444

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This volume contains the proceedings of the 19th International Conference on Difference Equations and Applications, held at Sultan Qaboos University, Muscat, Oman in May 2013. The conference brought together experts and novices in the theory and applications of difference equations and discrete dynamical systems. The volume features papers in difference equations and discrete time dynamical systems with applications to mathematical sciences and, in particular, mathematical biology, ecology, and epidemiology. It includes four invited papers and eight contributed papers. Topics covered include: competitive exclusion through discrete time models, Benford solutions of linear difference equations, chaos and wild chaos in Lorenz-type systems, advances in periodic difference equations, the periodic decomposition problem, dynamic selection systems and replicator equations, and asymptotic equivalence of difference equations in Banach Space. This book will appeal to researchers, scientists, and educators who work in the fields of difference equations, discrete time dynamical systems and their applications.

Advances in Difference Equations and Discrete Dynamical Systems

ICDEA, Osaka, Japan, July 2016

Author: Saber Elaydi,Yoshihiro Hamaya,Hideaki Matsunaga,Christian Pötzsche

Publisher: Springer

ISBN: 9811064091

Category: Mathematics

Page: 282

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This volume contains the proceedings of the 22nd International Conference on Difference Equations and Applications, held at Osaka Prefecture University, Osaka, Japan, in July 2016. The conference brought together both experts and novices in the theory and applications of difference equations and discrete dynamical systems. The volume features papers in difference equations and discrete dynamical systems with applications to mathematical sciences and, in particular, mathematical biology and economics. This book will appeal to researchers, scientists, and educators who work in the fields of difference equations, discrete dynamical systems, and their applications.

Dynamical Systems with Applications using MATLAB®

Author: Stephen Lynch

Publisher: Springer

ISBN: 3319068202

Category: Mathematics

Page: 514

View: 6733

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This textbook, now in its second edition, provides a broad introduction to both continuous and discrete dynamical systems, the theory of which is motivated by examples from a wide range of disciplines. It emphasizes applications and simulation utilizing MATLAB®, Simulink®, the Image Processing Toolbox® and the Symbolic Math toolbox®, including MuPAD. Features new to the second edition include · sections on series solutions of ordinary differential equations, perturbation methods, normal forms, Gröbner bases, and chaos synchronization; · chapters on image processing and binary oscillator computing; · hundreds of new illustrations, examples, and exercises with solutions; and · over eighty up-to-date MATLAB program files and Simulink model files available online. These files were voted MATLAB Central Pick of the Week in July 2013. The hands-on approach of Dynamical Systems with Applications using MATLAB, Second Edition, has minimal prerequisites, only requiring familiarity with ordinary differential equations. It will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a broad range of disciplines such as population dynamics, biology, chemistry, computing, economics, nonlinear optics, neural networks, and physics. Praise for the first edition Summing up, it can be said that this text allows the reader to have an easy and quick start to the huge field of dynamical systems theory. MATLAB/SIMULINK facilitate this approach under the aspect of learning by doing. —OR News/Operations Research Spectrum The MATLAB programs are kept as simple as possible and the author's experience has shown that this method of teaching using MATLAB works well with computer laboratory classes of small sizes.... I recommend ‘Dynamical Systems with Applications using MATLAB’ as a good handbook for a diverse readership: graduates and professionals in mathematics, physics, science and engineering. —Mathematica

Difference Equations, Discrete Dynamical Systems and Applications

ICDEA, Barcelona, Spain, July 2012

Author: Lluís Alsedà i Soler,Jim M. Cushing,Saber Elaydi,Alberto Adrego Pinto

Publisher: Springer

ISBN: 3662529270

Category: Mathematics

Page: 335

View: 8855

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These proceedings of the 18th International Conference on Difference Equations and Applications cover a number of different aspects of difference equations and discrete dynamical systems, as well as the interplay between difference equations and dynamical systems. The conference was organized by the Department of Mathematics at the Universitat Autònoma de Barcelona (UAB) under the auspices of the International Society of Difference Equations (ISDE) and held in Barcelona (Catalonia, Spain) in July 2012. Its purpose was to bring together experts and novices in these fields to discuss the latest developments. The book gathers contributions in the field of combinatorial and topological dynamics, complex dynamics, applications of difference equations to biology, chaotic linear dynamics, economic dynamics and control and asymptotic behavior, and periodicity of difference equations. As such it is of interest to researchers and scientists engaged in the theory and applications of difference equations and discrete dynamical systems.