Introduction to Chaos, Fractals and Dynamical Systems

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This book offers a fun and enriching introduction to chaos theory, fractals and dynamical systems, and on the applications of fractals to computer generated graphics and image compression. Introduction to Chaos, Fractals and Dynamical Systems particularly focuses on natural and human phenomenon that can be modeled as fractals, using simple examples to explain the theory of chaos and how it affects all of us. Then, using straightforward mathematic and intuitive descriptions, computer generated graphics and photographs of natural scenes are used to illustrate the beauty of fractals and their importance in our world. Finally, the concept of Dynamical Systems, that is, time-dependent systems, the foundation of Chaos and Fractal, is introduced. Everyday examples are again used to illustrate concepts, and the importance of understanding how these vital systems affect our lives. Throughout the fascinating history of the evolution of chaos theory, fractals and dynamical systems is presented, along with brief introductions to the scientists, mathematicians and engineers who created this knowledge. Introduction to Chaos, Fractals and Dynamical Systems contains ample mathematical definitions, representations, discussions and exercises, so that this book can be used as primary or secondary source in home schooling environments. The book is suitable for homeschooling as a focused course on the subject matter or as a classroom supplement for a variety of courses at the late junior high or early high-school level. For example, in addition to a standalone course on Chaos, Fractals and Dynamical Systems (or similar title), this book could be used with the following courses: Precalculus Geometry Computer programming (e.g. Rust, C, C++, Python, Java, Pascal) Computer graphics The text can also be used in conjunction with mathematics courses for undergraduates for non-science majors. The book can also be used for informal and lively family study and discussion. For each chapter, exercises and things to do are included. These activities range from simple computational tasks to more elaborate computer projects, related activities, biographical research and writing assignments. Sample Chapter(s) Introduction Chapter 1: What is Chaos? What are Fractals? Contents: What is Chaos? What are Fractals? Foundations of Chaos and Fractal Theory Chaos and Fractals in Nature Chaos and Fractals in Human-Made Phenomena Dynamical Systems and Systems Theory Readership: The market consists of mathematically inclined and/or homeschooled students from grades 6 or 7 through 12, and even early undergraduate, as well as computer science students the same grade levels as above. Hobbyists of all ages. Rust community — while this is not a book that intends to teach the Rust language, it is an impactful showcase of the language. We think we could generate some buzz for it on social media, for example by posting links to the program source code and full-color images (which are freely available online).

FRACTIONAL DISCRETE CHAOS

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In the nineteenth-century, fractional calculus had its origin in extending differentiation and integration operators from the integer-order case to the fractional-order case. Discrete fractional calculus has recently become an important research topic, useful in various science and engineering applications. The first definition of the fractional-order discrete-time/difference operator was introduced in 1974 by Diaz and Osler, where such operator was derived by discretizing the fractional-order continuous-time operator. Successfully, several types of fractional-order difference operators have then been proposed and introduced through further generalizing numerous classical operators, motivating several researchers to publish extensively on a new class of systems, viz the nonlinear fractional-order discrete-time systems (or simply, the fractional-order maps), and their chaotic behaviors. This discovery of chaos in such maps, has led to novel control methods for effectively stabilizing their chaotic dynamics. The aims of this book are as follows: Presenting the recent developments, trends, research solutions, applications and open problems related to fractional-order chaotic maps; Illustrating many interdisciplinary applications, like modulization, control, circuits, security and encryption; Including all theories associated with chaos, control and synchronization of discrete-time systems; Providing a useful reference on the topic of fractional-order chaotic maps and their applications. Sample Chapter(s) Preface Chapter 1: Discrete Fractional Calculus Contents: Discrete Fractional Calculus Chaotic Methods and Tests Chaos in 2D Discrete Fractional Systems Chaos in 3D Discrete Fractional Systems Applications of Fractional Chaotic Maps Readership: Graduate students and researchers in chaos, applied mathematics, computer science and engineering.

CLIMATE, CHAOS AND COVID

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Mathematical models are very much in the news now, as they are used to make decisions about our response to such vital areas as COVID-19 and climate change. Frequently, they are blamed for a series of dubious decisions, creating much concern amongst the general public. However, without mathematical models, we would have none of the modern technology that we take for granted, nor would we have modern health care, be able to forecast the climate, cook a potato, have electricity to power our home, or go into space. By explaining technical mathematical concepts in a way that everyone can understand and appreciate, Climate, Chaos and COVID: How Mathematical Models Describe the Universe sets the record straight and lifts the lid off the mystery of mathematical models. It shows why they work, how good they can be, the advantages and disadvantages of using them and how they make the modern world possible. The readers will be able to see the impact that the use of these models has on their lives, and will be able to appreciate both their power and their limitations. The book includes a very large number of both short and long case studies, many of which are taken directly from the author's own experiences of working as a mathematical modeller in academia, in industry, and between the two. These include COVID-19 and climate and how maths saves the whales, powers our home, gives us the material we need to live, and takes us into space. Sample Chapter(s) Preface Contents: Preface About the Author How Mathematics Models Real Life Why Did the Mathematician Cross the Road? Mathematics Saves the whales! How Mathematics Helped in the Fight Against COVID-19 Can Mathematics Predict the Future? The Mathematics of Climate Change Energetic Mathematics Mathematical Food for Thought Material Mathematics Mathematics Goes into Space References Index Readership: Keen high school students, teachers, STEM undergraduates and postgraduates interested in the field of applied mathematics and mathematical modelling; policymakers and decision makers in relevant industries such as Education, Environment, Health and Social Care, and Energy; members of the general public who want to understand more of the mathematics behind the models used in everyday life.