Kaosteori

The Federation of German Scientists together with the Institute of Medical Psychology, Ludwig Maximilian University, organized an interdisciplinary and international "autumn school" in 2023 when 30 young researchers and students participated. This important volume comprises lectures presented by international leading scientists and prominent experts from different academic fields who provided the background knowledge for "balanced sustainability in a changing world".Topics selected in the first block of lectures focused on climate change and biodiversity. And in the second block, lectures were given on social and personal challenges in a changing world: How should "smart cities" be organized in the future, and how a successful urban transformation can be managed. Then it was explained how a neuropolitical approach to humanization can help to overcome polarization in countries. The aspect of a necessary social togetherness was addressed in the third block with an example from Nepal and beyond. Finally, in a fourth block, questions of peace and intercultural spirit were evaluated.How can we reach and maintain equanimity in a changing world? How can peace be obtained? All these lectures were followed by intense discussions with the students, and their questions reflect the interdisciplinary background of the participants.It is hoped that the pragmatic suggestions from the participants of this autumn school will provide inspiration in finding solutions for imminent global and local problems.

This short work explores the excitement and surprising order of chaos and how studying it has revolutionized scientists' understanding of our world. ENDORSEMENTS: "Books on chaos theory tend to fall one of two ways. The first are those that cannot resist the hype. The exotic sound of "chaos," like "artificial intelligence," invites authors to lean into its mysteries. The second group of books is overly technical and beyond the reach of most readers. Physicists and mathematicians especially find it hard not to include all sorts of arcane detail. This book avoids both pitfalls. Robert Bishop deftly takes the reader through each of the key topics raised by chaos, including its historical roots, and shows why so many disciplines take note. Mathematicians provide fractal geometry as a tool. Computer scientists reveal how chaos puts limits on numerical simulations. Physical scientists have discovered that chaos is ubiquitous in nature. And philosophers explore its implications for a range of longstanding questions. With one foot in physics and another in the philosophy of science, Bishop illustrates each using minimal mathematics with important technical terms explained throughout. There is no better introduction to the topic available and no scholar better suited to the task. Jeffrey Koperski, Professor of Philosophy, Saginaw Valley State University "Sensitivity to initial conditions and path dependence, nonlinearities, strange attractors, and fractals...Bishop explains - in enviably clear language - the surprising properties and emergence of chaotic dynamics in mathematical models and real-world systems, and the differences between the two. The last chapter, on the limits chaos places on knowledge and predictability, provides insights on how individuals as well as scientists can work within those limits while at the same time embracing the remarkable wisdom they offer. Highly recommended. Alicia Juarrero, author of Context Changes Everything: How Constraints Create Coherence (MIT Press). "It won't come as a surprise to anyone who has been following Professor Bishop's work over the years on this topic that this book represents the best primer on chaos theory and nonlinear dynamics in existence today. Michael Silberstein, Professor of Philosophy, Elizabethtown College. "Chaos Theory: A Quick Immersion is a clear and engaging introduction to chaos theory. Assuming no prior knowledge, and using helpful analogies and examples from everyday life, it familiarizes readers with key concepts and findings of this fascinating field. The discussion is readily accessible to those new to the topic, yet without skirting over important nuances. In addition to surveying conceptual foundations, Chaos Theory nicely illustrates how ideas and tools from the mathematical study of chaos have been applied in science - in weather forecasting, ecology, physiology, physics, and more. Along the way, it calls attention to oft-overlooked challenges involved in relating mathematical and computer models to the physical world and emphasizes the importance of recognizing and navigating limits when seeking knowledge. This delightful little book will be useful to a wide range of readers interested in understanding what chaos theory is and how its insights can make a difference in science." Wendy Parker, Professor of Philosophy, Virginia Polytechnic University

Harnessing Circadian Rhythms for an Optimal Life" is a comprehensive guide that explores the fascinating world of circadian rhythms and how they can be utilized to enhance the quality of life. From understanding the science behind these internal clocks to practical strategies for aligning our routines with them, this book covers a wide range of topics. It delves into the impact of circadian rhythms on sleep, productivity, mental health, aging, travel, shift work, technology, and much more. Readers will discover actionable techniques for optimizing their daily routines, improving physical and cognitive performance, managing stress, and promoting overall well-being. With insights into the latest research and future possibilities, this book empowers individuals to embrace the power of circadian rhythms and unlock their full potential for a healthier, more fulfilling life.

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.

"The Art of Randomness teaches readers to harness the power of randomness (and Python code) to solve real-world problems in programming, science, and art through hands-on experiments-from simulating evolution to encrypting messages to making machine-learning algorithms. Each chapter describes how randomness plays into the given topic area, then proceeds to demonstrate its problem-solving role with hands-on experiments to work through using Python code"--

What kind of dynamics is a piecewise linear system able to display? How may they generate heteroclinic chaos? How can the coexistence of attractors be designed and characterized? Is it necessary to have equilibrium points to generate chaotic behavior? Chaos theory and complex systems are interesting and evolving topics whose investigation from a theoretical and practical point of view constantly leads to arising questions. Interesting behaviors can be observed in self-excited attractors, hidden attractors and non-self-excited attractors.This book presents some fundamentals of linear system theory and recent approaches to design the three classes of chaotic attractors in piecewise linear systems. Each chapter presents a brief description and basic concepts to provide an overview of linear systems theory; chaos and multistability in integer linear systems; hidden and non-self-excited attractors; and fractional approaches. They also provide example systems to illustrate the concepts and design methods introduced. Some current topics under investigation are addressed from an integer order perspective to make the connection with the fractional order counterpart.This textbook provides a comprehensive introduction, methodologies, and analysis tools to study chaotic piecewise linear systems and will be suitable for undergraduate or graduate students interested in the field of chaos and complex dynamics.

In the early 1980s, the late luminary Tito Arecchi was the first to highlight the existence of chaos in a laser model. Since then, along with several colleagues, he developed many important lines of research in this field, such as generalized multistability, laser with injected signal, laser with delayed feedback and the worldwide accepted classification of lasers of A, B and C, depending on their typical relaxation rates. Later, chaos control and synchronization were investigated in lasers and other systems, providing innovative schemes. Very recently, in his last contribution to laser physics, the model of the laser with feedback demonstrating its universal features was revisited.This book aims to present the research activity of Prof. Arecchi and his colleagues in the domain of nonlinear dynamics of lasers, since his seminal works of 1982 till the latest. Also included is our last contribution on jerk dynamics of laser's minimal universal model and a brief history of the discovery of laser where the reader will discover or rediscover many anecdotes about it.

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.

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 report describe encryp­tion in a wider class of computation, as compared to ordinary clas­sic encryption algorithms. The security of the pro­posed systems are investig­ated, and it is shown, that the non-al­gorithmic ciphers have many advant­ageous properties.The theoretical foundation is presented with many con­venient refer­ences. The report also contain a discussion of the relev­ance, of the theory, in the real world.Practical issues, of cipher design, are discussed, and the Reader should easily be able to design a secure cipher; adopted to any local requirements or restrictions.

This book provides a self-contained presentation of the physical and mathematical laws governing complex systems. Complex systems arising in natural, engineering, environmental, life and social sciences are approached from a unifying point of view using an array of methodologies such as microscopic and macroscopic level formulations, deterministic and probabilistic tools, modeling and simulation. The book can be used as a textbook by graduate students, researchers and teachers in science, as well as non-experts who wish to have an overview of one of the most open, markedly interdisciplinary and fast-growing branches of present-day science.

This book presents a comprehensive overview of the mathematics and physics behind the simulation of turbulent flows and discusses in detail (i) the phenomenology of turbulence in fluid dynamics, (ii) the role of direct and large-eddy simulation in predicting these dynamics, (iii) the multiple considerations underpinning subgrid modelling, and, (iv) the issue of validation and reliability resulting from interacting modelling and numerical errors.