Discrete Dynamical Systems

Discrete Dynamical Systems
Author: Oded Galor
Publsiher: Springer Science & Business Media
Total Pages: 153
Release: 2007-05-17
Genre: Business & Economics
ISBN: 9783540367765

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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

Discovering Discrete Dynamical Systems
Author: Aimee Johnson,Kathleen Madden,Ayse Sahin
Publsiher: American Mathematical Soc.
Total Pages: 116
Release: 2017-12-31
Genre: Mathematics
ISBN: 9781614441243

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Discovering Discrete Dynamical Systems is a mathematics textbook designed for use in a student-led, inquiry-based course for advanced mathematics majors. Fourteen modules each with an opening exploration, a short exposition and related exercises, and a concluding project guide students to self-discovery on topics such as fixed points and their classifications, chaos and fractals, Julia and Mandelbrot sets in the complex plane, and symbolic dynamics. Topics have been carefully chosen as a means for developing student persistence and skill in exploration, conjecture, and generalization while at the same time providing a coherent introduction to the fundamentals of discrete dynamical systems. This book is written for undergraduate students with the prerequisites for a first analysis course, and it can easily be used by any faculty member in a mathematics department, regardless of area of expertise. Each module starts with an exploration in which the students are asked an open-ended question. This allows the students to make discoveries which lead them to formulate the questions that will be addressed in the exposition and exercises of the module. The exposition is brief and has been written with the intent that a student who has taken, or is ready to take, a course in analysis can read the material independently. The exposition concludes with exercises which have been designed to both illustrate and explore in more depth the ideas covered in the exposition. Each module concludes with a project in which students bring the ideas from the module to bear on a more challenging or in-depth problem. A section entitled "To the Instructor" includes suggestions on how to structure a course in order to realize the inquiry-based intent of the book. The book has also been used successfully as the basis for an independent study course and as a supplementary text for an analysis course with traditional content.

A First Course in Discrete Dynamical Systems

A First Course in Discrete Dynamical Systems
Author: Richard A. Holmgren
Publsiher: Springer Science & Business Media
Total Pages: 223
Release: 2012-09-05
Genre: Mathematics
ISBN: 9781441987327

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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.

Introduction to Discrete Dynamical Systems and Chaos

Introduction to Discrete Dynamical Systems and Chaos
Author: Mario Martelli
Publsiher: John Wiley & Sons
Total Pages: 344
Release: 2011-11-01
Genre: Mathematics
ISBN: 9781118031124

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A timely, accessible introduction to the mathematics ofchaos. The past three decades have seen dramatic developments in thetheory of dynamical systems, particularly regarding the explorationof chaotic behavior. Complex patterns of even simple processesarising in biology, chemistry, physics, engineering, economics, anda host of other disciplines have been investigated, explained, andutilized. Introduction to Discrete Dynamical Systems and Chaos makes theseexciting and important ideas accessible to students and scientistsby assuming, as a background, only the standard undergraduatetraining in calculus and linear algebra. Chaos is introduced at theoutset and is then incorporated as an integral part of the theoryof discrete dynamical systems in one or more dimensions. Both phasespace and parameter space analysis are developed with ampleexercises, more than 100 figures, and important practical examplessuch as the dynamics of atmospheric changes and neuralnetworks. An appendix provides readers with clear guidelines on how to useMathematica 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 thatcan be assigned for classroom investigation. Based on the author's1993 book, but boasting at least 60% new, revised, and updatedmaterial, the present Introduction to Discrete Dynamical Systemsand Chaos is a unique and extremely useful resource for allscientists interested in this active and intensely studiedfield. An Instructor's Manual presenting detailed solutions to all theproblems in the book is available upon request from the Wileyeditorial department.

Discrete Dynamical Systems

Discrete Dynamical Systems
Author: James T. Sandefur
Publsiher: Oxford University Press, USA
Total Pages: 472
Release: 1990
Genre: Chaotic behavior in systems.
ISBN: UOM:39015062468114

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An elementary introduction to the world of dynamical systems and chaos. Dynamical systems provide a mathematical means of modelling and analyzing aspects of the changing world around us. The text aims to introduce both the techniques used in studying these systems and their applications.

Geometric Theory of Discrete Nonautonomous Dynamical Systems

Geometric Theory of Discrete Nonautonomous Dynamical Systems
Author: Christian Pötzsche
Publsiher: Springer
Total Pages: 399
Release: 2010-08-24
Genre: Mathematics
ISBN: 9783642142581

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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.

Formal Methods for Discrete Time Dynamical Systems

Formal Methods for Discrete Time Dynamical Systems
Author: Calin Belta,Boyan Yordanov,Ebru Aydin Gol
Publsiher: Springer
Total Pages: 284
Release: 2017-03-08
Genre: Technology & Engineering
ISBN: 9783319507637

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This book bridges fundamental gaps between control theory and formal methods. Although it focuses on discrete-time linear and piecewise affine systems, it also provides general frameworks for abstraction, analysis, and control of more general models. The book is self-contained, and while some mathematical knowledge is necessary, readers are not expected to have a background in formal methods or control theory. It rigorously defines concepts from formal methods, such as transition systems, temporal logics, model checking and synthesis. It then links these to the infinite state dynamical systems through abstractions that are intuitive and only require basic convex-analysis and control-theory terminology, which is provided in the appendix. Several examples and illustrations help readers understand and visualize the concepts introduced throughout the book.

An Introduction to Dynamical Systems

An Introduction to Dynamical Systems
Author: Rex Clark Robinson
Publsiher: American Mathematical Soc.
Total Pages: 763
Release: 2012
Genre: Mathematics
ISBN: 9780821891353

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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.