Discovering Discrete Dynamical Systems

Discovering Discrete Dynamical Systems
Author: Aimee Johnson
Publisher: American Mathematical Soc.
Total Pages: 132
Release: 2017-12-31
Genre: Mathematics
ISBN: 0883857936
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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.

Discovering Discrete Dynamical Systems

Discovering Discrete Dynamical Systems
Author: Aimee Johnson
Publisher: American Mathematical Soc.
Total Pages: 116
Release: 2017-12-31
Genre: Mathematics
ISBN: 1614441243
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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
Publisher: Springer Science & Business Media
Total Pages: 231
Release: 2012-09-05
Genre: Mathematics
ISBN: 1441987320
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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

Discrete Dynamical Systems
Author: Oded Galor
Publisher: Springer Science & Business Media
Total Pages: 159
Release: 2007-05-17
Genre: Business & Economics
ISBN: 3540367764
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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.

Chaos in Discrete Dynamical Systems

Chaos in Discrete Dynamical Systems
Author: Ralph Abraham
Publisher: Springer Science & Business Media
Total Pages: 257
Release: 2013-06-29
Genre: Mathematics
ISBN: 1461219361
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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.

Discovering Dynamical Systems Through Experiment and Inquiry

Discovering Dynamical Systems Through Experiment and Inquiry
Author: Thomas LoFaro
Publisher: CRC Press
Total Pages: 216
Release: 2021-03-22
Genre: Mathematics
ISBN: 1000358933
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Discovering Dynamical Systems Through Experiment and Inquiry differs from most texts on dynamical systems by blending the use of computer simulations with inquiry-based learning (IBL). IBL is an excellent tool to move students from merely remembering the material to deeper understanding and analysis. This method relies on asking students questions first, rather than presenting the material in a lecture. Another unique feature of this book is the use of computer simulations. Students can discover examples and counterexamples through manipulations built into the software. These tools have long been used in the study of dynamical systems to visualize chaotic behavior. We refer to this unique approach to teaching mathematics as ECAP—Explore, Conjecture, Apply, and Prove. ECAP was developed to mimic the actual practice of mathematics in an effort to provide students with a more holistic mathematical experience. In general, each section begins with exercises guiding students through explorations of the featured concept and concludes with exercises that help the students formally prove the results. While symbolic dynamics is a standard topic in an undergraduate dynamics text, we have tried to emphasize it in a way that is more detailed and inclusive than is typically the case. Finally, we have chosen to include multiple sections on important ideas from analysis and topology independent from their application to dynamics.

Introduction to Discrete Dynamical Systems and Chaos

Introduction to Discrete Dynamical Systems and Chaos
Author: Mario Martelli
Publisher: John Wiley & Sons
Total Pages: 347
Release: 2011-11-01
Genre: Mathematics
ISBN: 1118031121
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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.

Formal Methods for Discrete-Time Dynamical Systems

Formal Methods for Discrete-Time Dynamical Systems
Author: Calin Belta
Publisher: Springer
Total Pages: 291
Release: 2017-03-08
Genre: Technology & Engineering
ISBN: 331950763X
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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.

Invitation to Dynamical Systems

Invitation to Dynamical Systems
Author: Edward R. Scheinerman
Publisher: Courier Corporation
Total Pages: 402
Release: 2012-01-01
Genre: Mathematics
ISBN: 0486485943
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This text is designed for those who wish to study mathematics beyond linear algebra but are not ready for abstract material. Rather than a theorem-proof-corollary-remark style of exposition, it stresses geometry, intuition, and dynamical systems. An appendix explains how to write MATLAB, Mathematica, and C programs to compute dynamical systems. 1996 edition.

Exploring Discrete Dynamics. 2nd Editiion. the Ddlab Manual

Exploring Discrete Dynamics. 2nd Editiion. the Ddlab Manual
Author: Andrew Wuensche
Publisher:
Total Pages: 578
Release: 2016-06-15
Genre: Computers
ISBN: 9781905986477
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EXPLORING DISCRETE DYNAMICS (second edition) is a comprehensive guide to studying cellular automata and discrete dynamical networks with the classic software Discrete Dynamics Laboratory (DDLab), widely used in research and education. These collective networks are at the core of complexity and emergent self-organisation. With interactive graphics, DDLab is able to explore a huge diversity of behaviour, mostly terra incognita -- space-time patterns, and basins of attraction -- mathematical objects representing the convergent flow in state-space. Applications range within physics, mathematics, biology, cognition, society, economics and computation, and more specifically in neural and genetic networks, artificial life, and theories of memory. This second edition covers many new features. Advance Praise by Stuart Kauffman The great John von Neumann invented cellular automata. These discrete state finite automata have become a mainstay in the study of complex systems, exhibiting order, criticality, and chaos. Andy Wuensche's "Exploring Discrete Dynamics" 2016, is by far the most advanced tool for simulating such systems and has become widely important in the field of complexity. FIRST EDITION REVIEWS Andrew Wuensche has, in an important sense, done more than anyone to enable the study of discrete dynamical systems such as cellular automata and random Boolean nets. Wuensche derived the mathematical means to compute the "predecessor" states that flow to a successor state. Thereby he opened the door to study the entire state space flow of discrete dynamical systems. DDLab is a marvellous and useful tool for all of us fascinated by discrete dynamical systems and what they may tell us of mathematics and the world. STUART KAUFFMAN, author of "The Origins of Order" Tampere University of Technology, Finland. There is a whole universe of complexity that is captured by discrete dynamical systems, which have been widely used as a powerful framework to understand reality from different perspectives. Exploring Discrete Dynamics is a great example of how to dive in this neverending universe. A careful, compelling and detailed presentation of examples and methods will help both beginners and scholars to get into this fascinating field." RICARD SOLE, Author of "Signs of Life" Complex Systems Lab, Universitat Pompeu Fabra, Barcelona.