Exploring Discrete Dynamics

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.

EXPLORING DISCRETE DYNAMICS is a comprehensive guide to studying cellular automata and discrete dynamical networks with the classic software Discrete Dynamics Laboratory (DDLab). These collective networks are at the core of complexity and emergent self-organisation. With interactive graphics, DDLab is able to explore an huge diversity of behaviour -- mostly terra incognita -- space-time patters, but also 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 a theory of memory.

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 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 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.

This book is conceived as a comprehensive and detailed text-book on non-linear dynamical systems with particular emphasis on the exploration of chaotic phenomena. The self-contained introductory presentation is addressed both to those who wish to study the physics of chaotic systems and non-linear dynamics intensively as well as those who are curious to learn more about the fascinating world of chaotic phenomena. Basic concepts like Poincaré section, iterated mappings, Hamiltonian chaos and KAM theory, strange attractors, fractal dimensions, Lyapunov exponents, bifurcation theory, self-similarity and renormalisation and transitions to chaos are thoroughly explained. To facilitate comprehension, mathematical concepts and tools are introduced in short sub-sections. The text is supported by numerous computer experiments and a multitude of graphical illustrations and colour plates emphasising the geometrical and topological characteristics of the underlying dynamics. This volume is a completely revised and enlarged second edition which comprises recently obtained research results of topical interest, and has been extended to include a new section on the basic concepts of probability theory. A completely new chapter on fully developed turbulence presents the successes of chaos theory, its limitations as well as future trends in the development of complex spatio-temporal structures. "This book will be of valuable help for my lectures" Hermann Haken, Stuttgart "This text-book should not be missing in any introductory lecture on non-linear systems and deterministic chaos" Wolfgang Kinzel, Würzburg “This well written book represents a comprehensive treatise on dynamical systems. It may serve as reference book for the whole field of nonlinear and chaotic systems and reports in a unique way on scientific developments of recent decades as well as important applications.” Joachim Peinke, Institute of Physics, Carl-von-Ossietzky University Oldenburg, Germany

The book reviews the application of discrete fractional operators in diverse fields such as biological and chemical reactions, as well as chaotic systems, demonstrating their applications in physics. The dynamical analysis is carried out using equilibrium points of the system for studying their stability properties and the chaotic behaviors are illustrated with the help of bifurcation diagrams and Lyapunov exponents. The book is divided into three parts. Part I deals with the application of discrete fractional operators in chemical reaction-based systems with biological significance. Two different chemical reaction models are analysed- one being disproportionation of glucose, which plays an important role in human physiology and the other is the Lengyel – Epstein chemical model. Chaotic behavior of the systems is studied and the synchronization of the system is performed. Part II covers the analysis of biological systems like tumor immune system and neuronal models by introducing memristor based flux control. The memductance functions are considered as quadratic, periodic, and exponential functions. The final part of the book reviews the complex form of the Rabinovich-Fabrikant system which describes physical systems with strong nonlinearity exhibiting unusual behavior.

This volume, with a Foreword writer Sir Roger Penrose, discusses the foundations of computation in relation to nature. It focuses on two main questions: What is computation?How does nature compute? The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and philosophical implications. The volume provides a state-of-the-art collection of technical papers and non-technical essays, representing a field that assumes information and computation to be key in understanding and explaining the basic structure underpinning physical reality. It also includes a new edition of Konrad Zuse's “Calculating Space” (the MIT translation), and a panel discussion transcription on the topic, featuring worldwide experts in quantum mechanics, physics, cognition, computation and algorithmic complexity. The volume is dedicated to the memory of Alan M Turing — the inventor of universal computation, on the 100th anniversary of his birth, and is part of the Turing Centenary celebrations. Contents:Foreword (R Penrose)PrefaceAcknowledgementsIntroducing the Computable Universe (H Zenil)Historical, Philosophical & Foundational Aspects of Computation:Origins of Digital Computing: Alan Turing, Charles Babbage, & Ada Lovelace (D Swade)Generating, Solving and the Mathematics of Homo Sapiens. E Post's Views on Computation (L De Mol)Machines (R Turner)Effectiveness (N Dershowitz & E Falkovich)Axioms for Computability: Do They Allow a Proof of Church's Thesis? (W Sieg)The Mathematician's Bias — and the Return to Embodied Computation (S B Cooper)Intuitionistic Mathematics and Realizability in the Physical World (A Bauer)What is Computation? Actor Model versus Turing's Model (C Hewitt)Computation in Nature & the Real World:Reaction Systems: A Natural Computing Approach to the Functioning of Living Cells (A Ehrenfeucht, J Kleijn, M Koutny & G Rozenberg)Bacteria, Turing Machines and Hyperbolic Cellular Automata (M Margenstern)Computation and Communication in Unorganized Systems (C Teuscher)The Many Forms of Amorphous Computational Systems (J Wiedermann)Computing on Rings (G J Martínez, A Adamatzky & H V McIntosh)Life as Evolving Software (G J Chaitin)Computability and Algorithmic Complexity in Economics (K V Velupillai & S Zambelli)Blueprint for a Hypercomputer (F A Doria)Computation & Physics & the Physics of Computation:Information-Theoretic Teleodynamics in Natural and Artificial Systems (A F Beavers & C D Harrison)Discrete Theoretical Processes (DTP) (E Fredkin)The Fastest Way of Computing All Universes (J Schmidhuber)The Subjective Computable Universe (M Hutter)What Is Ultimately Possible in Physics? (S Wolfram)Universality, Turing Incompleteness and Observers (K Sutner)Algorithmic Causal Sets for a Computational Spacetime (T Bolognesi)The Computable Universe Hypothesis (M P Szudzik)The Universe is Lawless or “Pantôn chrêmatôn metron anthrôpon einai” (C S Calude, F W Meyerstein & A Salomaa)Is Feasibility in Physics Limited by Fantasy Alone? (C S Calude & K Svozil)The Quantum, Computation & Information:What is Computation? (How) Does Nature Compute? (D Deutsch)The Universe as Quantum Computer (S Lloyd)Quantum Speedup and Temporal Inequalities for Sequential Actions (M Żukowski)The Contextual Computer (A Cabello)A Gödel-Turing Perspective on Quantum States Indistinguishable from Inside (T Breuer)When Humans Do Compute Quantum (P Zizzi)Open Discussion Section:Open Discussion on A Computable Universe (A Bauer, T Bolognesi, A Cabello, C S Calude, L De Mol, F Doria, E Fredkin, C Hewitt, M Hutter, M Margenstern, K Svozil, M Szudzik, C Teuscher, S Wolfram & H Zenil)Live Panel Discussion (transcription):What is Computation? (How) Does Nature Compute? (C S Calude, G J Chaitin, E Fredkin, A J Leggett, R de Ruyter, T Toffoli & S Wolfram)Zuse's Calculating Space:Calculating Space (Rechnender Raum) (K Zuse)Afterword to Konrad Zuse's Calculating Space (A German & H Zenil) Readership: Graduate students who are specialized researchers in computer science, information theory, quantum theory and modern philosophy and the general public who are interested in these subject areas. Keywords:Digital Physics;Computational Universe;Digital Philosophy;Reality Theories of the Universe;Models of the World;Thring Computation RandomnessKey Features:The authors are all prominent researchersNo competing titlesState-of-the-art collection of technical papers and non-technical essays