Julien Clinton Sprott - Böcker

This text provides an introduction to the exciting new developments in chaos and related topics in nonlinear dynamics, including the detection and quantification of chaos in experimental data, fractals, and complex systems. Most of the important elementary concepts in nonlinear dynamics are discussed, with emphasis on the physical concepts and useful results rather than mathematical proofs and derivations. While many books on chaos are purely qualitative and many others are highly mathematical, this book fills the middle ground by giving the essential equations, but in the simplest possible form. It assumes only an elementary knowledge of calculus. Complex numbers, differential equations, and vector calculus are used in places, but those tools are described as required. The book is aimed at the student, scientist, or engineer who wants to learn how to use the ideas in a practical setting. It is written at a level suitable for advanced undergraduate and beginning graduate students in all fields of science and engineering.

An introduction to developments in chaos and related topics in nonlinear dynamics, including the detection and quantification of chaos in experimental data, fractals, and complex systems. Most of the important elementary concepts in nonlinear dynamics are discussed, with emphasis on the physical concepts and useful results rather than mathematical proofs and derivations. While many books on chaos are purely qualitative and many others are highly mathematical, this book fills the middle ground by giving the essential equations, but in the simplest possible form. It assumes only an elementary knowledge of calculus. Complex numbers, differential equations, and vector calculus are used in places, but those tools are described as required. The book is aimed at the student, scientist, or engineer who wants to learn how to use the ideas in a practical setting. It is written at a level suitable for advanced undergraduate and beginning graduate students in all fields of science and engineering.

Chaos is the study of the underlying determinism in the seemingly random phenomena that occur all around us. One of the best experimental demonstrations of chaos occurs in electrical circuits when the parameters are chosen carefully. We will show you how to construct such chaotic circuits for use in your own studies and demonstrations while teaching you the basics of chaos.This book should be of interest to researchers and hobbyists looking for a simple way to produce a chaotic signal. It should also be useful to students and their instructors as an engaging way to learn about chaotic dynamics and electronic circuits. The book assumes only an elementary knowledge of calculus and the ability to understand a schematic diagram and the components that it contains.You will get the most out of this book if you can construct the circuits for yourself. There is no substitute for the thrill and insight of seeing the output of a circuit you built unfold as the trajectory wanders in real time across your oscilloscope screen. A goal of this book is to inspire and delight as well as to teach.

A recent development is the discovery that simple systems of equations can have chaotic solutions in which small changes in initial conditions have a large effect on the outcome, rendering the corresponding experiments effectively irreproducible and unpredictable. An earlier book in this sequence, Elegant Chaos: Algebraically Simple Chaotic Flows provided several hundred examples of such systems, nearly all of which are purely mathematical without any obvious connection with actual physical processes and with very limited discussion and analysis.In this book, we focus on a much smaller subset of such models, chosen because they simulate some common or important physical phenomenon, usually involving the motion of a limited number of point-like particles, and we discuss these models in much greater detail. As with the earlier book, the chosen models are the mathematically simplest formulations that exhibit the phenomena of interest, and thus they are what we consider 'elegant.'Elegant models, stripped of unnecessary detail while maximizing clarity, beauty, and simplicity, occupy common ground bordering both real-world modeling and aesthetic mathematical analyses. A computational search led one of us (JCS) to the same set of differential equations previously used by the other (WGH) to connect the classical dynamics of Newton and Hamilton to macroscopic thermodynamics. This joint book displays and explores dozens of such relatively simple models meeting the criteria of elegance, taste, and beauty in structure, style, and consequence.This book should be of interest to students and researchers who enjoy simulating and studying complex particle motions with unusual dynamical behaviors. The book assumes only an elementary knowledge of calculus. The systems are initial-value iterated maps and ordinary differential equations but they must be solved numerically. Thus for readers a formal differential equations course is not at all necessary, of little value and limited use.

This book was mostly written by a machine that was programmed to search a system of equations for chaotic solutions, simplify the equations to the extent possible, analyze the behavior, produce figures, and write the accompanying text. The equations are coupled autonomous ordinary differential equations with three variables and at least one nonlinearity. Fifty simple systems are included. Some are old and familiar; others are relatively new and unknown. They are chosen to illustrate by simple example most of dynamical behaviors that can occur in low-dimensional chaotic systems.There is no substitute for the thrill and insight of seeing the solution of a simple equation unfold as the trajectory wanders in real time across your computer screen using a program of your own making. A goal of this book is to inspire and delight as well as to teach. It provides a wealth of examples ripe for further study and extension, and it offers a glimpse of a future when artificial intelligence supplants many of the mundane tasks that accompany dynamical systems research and becomes a true and tireless collaborator.

With the poems written by winner of the Posner Poetry Award from the Council of Wisconsin Writers in 2005, this coffee-table book will delight and inform general readers curious about ideas of chaos, fractals, and nonlinear complex systems. Developed out of ten years of interdisciplinary seminars in chaos and complex systems at the University of Wisconsin-Madison, it features multiple ways of knowing: Robin Chapman's poems of everyday experience of change in a complex world, associated metaphorically with Julien Clinton Sprott's full-color computer art generated from billions of versions of only three simple equations for strange attractors, Julia sets, and iterated function systems; his definitions of 39 key terms; a mathematical appendix; and even a multiple-choice quiz to test understanding. Accompanied by a CD-ROM of the poet reading 13 poems and 1,000 images of chaos art from which slide shows can be generated and 100 high-resolution posters created, the book has a foreword by Cliff Pickover, author of A Passion for Mathematics.

With the poems written by winner of the Posner Poetry Award from the Council of Wisconsin Writers in 2005, this coffee-table book will delight and inform general readers curious about ideas of chaos, fractals, and nonlinear complex systems. Developed out of ten years of interdisciplinary seminars in chaos and complex systems at the University of Wisconsin-Madison, it features multiple ways of knowing: Robin Chapman's poems of everyday experience of change in a complex world, associated metaphorically with Julien Clinton Sprott's full-color computer art generated from billions of versions of only three simple equations for strange attractors, Julia sets, and iterated function systems; his definitions of 39 key terms; a mathematical appendix; and even a multiple-choice quiz to test understanding. Accompanied by a CD-ROM of the poet reading 13 poems and 1,000 images of chaos art from which slide shows can be generated and 100 high-resolution posters created, the book has a foreword by Cliff Pickover, author of A Passion for Mathematics.

This heavily illustrated book collects in one source most of the mathematically simple systems of differential equations whose solutions are chaotic. It includes the historically important systems of van der Pol, Duffing, Ueda, Lorenz, Rössler, and many others, but it goes on to show that there are many other systems that are simpler and more elegant. Many of these systems have been only recently discovered and are not widely known. Most cases include plots of the attractor and calculations of the spectra of Lyapunov exponents. Some important cases include graphs showing the route to chaos. The book includes many cases not previously published as well as examples of simple electronic circuits that exhibit chaos.No existing book thus far focuses on mathematically elegant chaotic systems. This book should therefore be of interest to chaos researchers looking for simple systems to use in their studies, to instructors who want examples to teach and motivate students, and to students doing independent study.

This heavily illustrated book collects in one source most of the mathematically simple systems of differential equations whose solutions are chaotic. It includes the historically important systems of van der Pol, Duffing, Ueda, Lorenz, Rössler, and many others, but it goes on to show that there are many other systems that are simpler and more elegant. Many of these systems have been only recently discovered and are not widely known. Most cases include plots of the attractor and calculations of the spectra of Lyapunov exponents. Some important cases include graphs showing the route to chaos. The book includes many cases not previously published as well as examples of simple electronic circuits that exhibit chaos.No existing book thus far focuses on mathematically elegant chaotic systems. This book should therefore be of interest to chaos researchers looking for simple systems to use in their studies, to instructors who want examples to teach and motivate students, and to students doing independent study.

Fractals are intricate geometrical forms that contain miniature copies of themselves on ever smaller scales. This colorful book describes methods for producing an endless variety of fractal art using a computer program that searches through millions of equations looking for those few that can produce images having aesthetic appeal. Over a hundred examples of such images are included with a link to the software that produced these images, and can also produce many more similar fractals. The underlying mathematics of the process is also explained in detail.Other books by the author that could be of interest to the reader are Elegant Chaos: Algebraically Simple Chaotic Flows (J C Sprott, 2010) and Elegant Circuits: Simple Chaotic Oscillators (J C Sprott and W J Thio, 2020).

This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hénon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters.Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hénon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincaré map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincaré mapping in addition to other analytical methods.

This collection of review articles is devoted to new developments in the study of chaotic dynamical systems with some open problems and challenges. The papers, written by many of the leading experts in the field, cover both the experimental and theoretical aspects of the subject. This edited volume presents a variety of fascinating topics of current interest and problems arising in the study of both discrete and continuous time chaotic dynamical systems. Exciting new techniques stemming from the area of nonlinear dynamical systems theory are currently being developed to meet these challenges. Presenting the state-of-the-art of the more advanced studies of chaotic dynamical systems, Frontiers in the Study of Chaotic Dynamical Systems with Open Problems is devoted to setting an agenda for future research in this exciting and challenging field.