Fractals and Chaos: An Illustrated Course provides you with a practical, elementary introduction to fractal geometry and chaotic dynamics-subjects that have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. In addition, plenty of problems are provided to test understanding. Advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study.

When a dynamical system has a large number of parameters it is not possible to get a completely comprehensive picture of all the types of behavior that it may display and one must be content with surveying the system along various corridors of lower dimension. Using an example with three differential equations and six parameters it is shown how the available methods of singularity theory, bifurcation analysis, normal forms, etc. can be used to build up a picture of varied and interesting behavior. The model is a generalization of the Gray-Scott reaction scheme in a single stirred vessel to a two-phase reactor consisting of a reaction chamber and a reservoir communicating with each other through a semi-permeable membrane. Two forms exist according as to whether A is fed to the reactor and B to the reservoir or vice-versa, and show interesting differences of behavior. Both models undergo Hopf bifurcations, pitchfork transitions, have homoclinic orbits, take the period doubling route to chaos and one gets there by intermittency. Besides being of interest to mathematicians as an ecological study of a differentiable system, it is hoped that, though idealized, the fact that it corresponds closely to a real type of reactor will make it attractive to control engineers and others as a testing ground for their various methods and devices. This book will be of particular interest to students and researchers in mathematics and engineering , particularly those working in bifurcation or chaos theory.

Nonlinear Dynamics: A Two-Way Trip from Physics to Math provides readers with the mathematical tools of nonlinear dynamics to tackle problems in all areas of physics. The selection of topics emphasizes bifurcation theory and topological analysis of dynamical systems. The book includes real-life problems and experiments as well as exercises and worked examples to test understanding.

This book represents the best of the first three years of the Society for Chaos Theory in Psychology conferences. While chaos theory has been a topic of considerable interest in the physical and biological sciences, its applications in psychology and related fields have been obscured until recently by its complexity. Nevertheless, a small but rapidly growing community of psychologists, neurobiologists, sociologists, mathematicians, and philosophers have been coming together to discuss its implications and explore its research possibilities.
Chaos theory has been termed the first authentic paradigm shift since the advent of quantum physics. Whether this is true or not, it unquestionably bears profound implications for many fields of thought. These include the cognitive analysis of the mind, the nature of personality, the dynamics of psychotherapy and counseling, understanding brain events and behavioral records, the dynamics of social organization, and the psychology of prediction. To each of these topics, chaos theory brings the perspective of dynamic self-organizing processes of exquisite complexity. Behavior, the nervous system, and social processes exhibit many of the classical characteristics of chaotic systems -- they are deterministic and globally predictable and yet do not submit to precise predictability.
This volume is the first to explore ideas from chaos theory in a broad, psychological perspective. Its introduction, by the prominent neuroscientist Walter Freeman, sets the tone for diverse discussions of the role of chaos theory in behavioral research, the study of personality, psychotherapy and counseling, mathematical cognitive psychology, social organization, systems philosophy, and the understanding of the brain.

The discipline of nonlinear dynamics has developed explosively in all areas of physics. This comprehensive primer summarizes the main developments in the mathematical theory of dynamical systems, chaos, pattern formation and complexity. An introduction to mathematical concepts and techniques is given in the first part of the book, before being applied to stellar, interstellar, galactic and large scale complex phenomena in the Universe. Regev demonstrates the possible application of ideas including strange attractors, Poincare sections, fractals, bifurcations, and complex spatial patterns, to specific astrophysical problems. This self-contained text will appeal to a broad audience of astrophysicists and astronomers who wish to understand and apply modern dynamical approaches to the problems they are working on. It provides researchers and graduate students with the investigative tools they need to fully explore chaotic and complex phenomena.

This book deals with the bifurcation and chaotic aspects of damped and driven nonlinear oscillators. The analytical and numerical aspects of the chaotic dynamics of these oscillators are covered, together with appropriate experimental studies using nonlinear electronic circuits. Recent exciting developments in chaos research are also discussed, such as the control and synchronization of chaos and possible technological applications.

Presents new computer methods in approximation, simulation, and visualization for a host of alpha-stable stochastic processes.

Accurate predictions of storm surge are of importance in many coastal areas in the world to avoid and mitigate its destructive impacts. For this purpose the physically-based (process) numerical models are typically utilized. However, in data-rich cases, one may use data-driven methods aiming at reconstructing the internal patterns of the modelled processes and relationships between the observed descriptive variables. This book focuses on data-driven modelling using methods of nonlinear dynamics and chaos theory. First, some fundamentals of physical oceanography, nonlinear dynamics and chaos, computational intelligence and European operational storm surge models are covered. After that a number of improvements in building chaotic models are presented: nonlinear time series analysis, multi-step prediction, phase space dimensionality reduction, techniques dealing with incomplete time series, phase error correction, finding true neighbours, optimization of chaotic model, data assimilation and multi-model ensemble prediction. The major case study is surge prediction in the North Sea, with some tests on a Caribbean Sea case. The modelling results showed that the enhanced predictive chaotic models can serve as an efficient tool for accurate and reliable short and mid-term predictions of storm surges in order to support decision-makers for flood prediction and ship navigation.

The study of nonlinear dynamical systems has been gathering momentum since the late 1950s. It now constitutes one of the major research areas of modern theoretical physics. The twin themes of fractals and chaos, which are linked by attracting sets in chaotic systems that are fractal in structure, are currently generating a great deal of excitement. The degree of structure robustness in the presence of stochastic and quantum noise is thus a topic of interest. Chaos, Noise and Fractals discusses the role of fractals in quantum mechanics, the influence of phase noise in chaos and driven optical systems, and the arithmetic of chaos. The book represents a balanced overview of the field and is a worthy addition to the reading lists of researchers and students interested in any of the varied, and sometimes bizarre, aspects of this intriguing subject.

While many books have discussed methodological advances in nonlinear dynamical systems theory (NDS), this volume is unique in its focus on NDS s role in the development of psychological theory. After an introductory chapter covering the fundamentals of chaos, complexity, and other nonlinear dynamics, subsequent chapters provide in-depth coverage of each of the specific topic areas in psychology. A concluding chapter takes stock of the field as a whole, evaluating important challenges for the immediate future. The chapters are written by experts in the use of NDS in each of their respective areas, including biological, cognitive, developmental, social, organizational, and clinical psychology. Each chapter provides an in-depth examination of theoretical foundations and specific applications and a review of relevant methods. This edited collection represents the state of the art in NDS science across the disciplines of psychology."

Accurate predictions of storm surge are of importance in many coastal areas in the world to avoid and mitigate its destructive impacts. For this purpose the physically-based (process) numerical models are typically utilized. However, in data-rich cases, one may use data-driven methods aiming at reconstructing the internal patterns of the modelled processes and relationships between the observed descriptive variables. This book focuses on data-driven modelling using methods of nonlinear dynamics and chaos theory. First, some fundamentals of physical oceanography, nonlinear dynamics and chaos, computational intelligence and European operational storm surge models are covered. After that a number of improvements in building chaotic models are presented: nonlinear time series analysis, multi-step prediction, phase space dimensionality reduction, techniques dealing with incomplete time series, phase error correction, finding true neighbours, optimization of chaotic model, data assimilation and multi-model ensemble prediction. The major case study is surge prediction in the North Sea, with some tests on a Caribbean Sea case. The modelling results showed that the enhanced predictive chaotic models can serve as an efficient tool for accurate and reliable short and mid-term predictions of storm surges in order to support decision-makers for flood prediction and ship navigation.

This book presents a select group of papers that provide a comprehensive view of the models and applications of chaos theory in medicine, biology, ecology, economy, electronics, mechanical, and the human sciences. Covering both the experimental and theoretical aspects of the subject, it examines a range of current topics of interest. It considers the problems arising in the study of discrete and continuous time chaotic dynamical systems modeling the several phenomena in nature and society highlighting powerful techniques being developed to meet these challenges that stem from the area of nonlinear dynamical systems theory.

This volume provides a self-contained survey of the mechanisms presiding information processing and communication. The main thesis is that chaos and complexity are the basic ingredients allowing systems composed of interesting subunits to generate and process information and communicate in a meaningful way. Emphasis is placed on communication in the form of games and on the related issue of decision making under conditions of uncertainty. Biological, cognitive, physical, engineering and societal systems are approached from a unifying point of view, both analytically and by numerical simulation, using the methods of nonlinear dynamics and probability theory. Epistemological issues in connection with incompleteness and self-reference are also addressed.

The book is a compilation of selected papers from the conference on Physics and Control 2009, presenting a unified perspective underlying the thematics and strategies related to the control of physical systems with emerging applications in physics, engineering, chemistry, biology and other natural sciences. The selected papers reflect the state-of-the-art of the more advanced theoretical and practical studies in the field of control of complex systems. The contributions provide a comprehensive view on some selected topics of particular importance at the disciplinary borderline between Physics and Control.

This self-contained textbook covers the fundamentals of two basic topics of linear functional analysis: locally convex spaces and harmonic analysis. Readers will find detailed introductions to topological vector spaces, distribution theory, weak topologies, Fourier transform, Hilbert transform, and Calderon-Zygmund singular integrals. An ideal introduction to more advanced texts, the book complements Ciarlet's Linear and Nonlinear Functional Analysis with Applications (SIAM), where the titular topics were not treated. Pedagogical features such as detailed theorems and 93 problems make the book ideal for a one-semester first-year graduate course or for self-study. The book is intended for advanced undergraduates and first-year graduate students and researchers. It is appropriate for the following courses: Advanced Numerical Analysis, Special Topics on Numerical Analysis, Topics on Data Science, Topics on Numerical Optimization, and Topics on Approximation Theory

This book, the first in the Cambridge Nonlinear Science Series, presents the fundamentals of chaos theory in conservative systems, providing a systematic study of the theory of transitional states of physical systems which lie between deterministic and chaotic behaviour. The authors' treatment of transitions to chaos, the theory of stochastic layers and webs, and the numerous applications of this theory, particularly to pattern symmetry, will make the book of importance to scientists from many disciplines.The authors have been meticulous in providing a detailed presentation of the material, enabling the reader to learn the necessary computational methods and to apply them in other problems. The inclusion of a significant amount of computer graphics is also an important aid to understanding. The final section of the book contains a fascinating collection of patterns in art and living nature. The book will be of interest to graduate students and researchers in physics and mathematics who are interested in problems of chaos, irreversibility, statistical mechanics and theories of spatial patterns and symmetries. The perhaps unconventional links between chaos theory and other topics make the book particularly interesting.

One of the most penetrating and celebrated thinkers writing about the philosophy of science today, Isabelle Stengers here provides a firsthand account of the meeting of science and history. Concerned with the force and inventiveness of those theories, Power and Invention offers a unique perspective on the power of scientific theories to modify society, and vice versa.

Using the law of thermodynamics, Stengers sets out to explain the consequences of nonlinear dynamics (or chaos theory) for philosophy and science. She makes a case for the concept of complexity that transcends the conventional boundaries of scientific discourse and that clearly exposes the risks of scientific theories. Among the questions she confronts are: Is psychoanalysis a science? Is there such a thing as "women's science"? What are scientific theories?

Self-organization of matter is observed in every context and on all scales, from the nanoscale of quantum fields and subatomic particles to the macroscale of galaxy superclusters. This book analyzes the wide range of patterns of organization present in nature, highlighting their similarities rather than their differences. This unconventional approach results in an illuminating read which should be part of any Physics student's background.

Nonlinear Dynamics, Volume 1. Proceedings of the 34th IMAC, A Conference and Exposition on Dynamics of Multiphysical Systems: From Active Materials to Vibroacoustics, 2016, the fi rst volume of ten from the Conference, brings together contributions to this important area of research and engineering. Th e collection presents early fi ndings and case studies on fundamental and applied aspects of Structural Dynamics, including papers on: * Nonlinear Oscillations * Nonlinear Modal Analysis * Nonlinear System Identifi cation * Nonlinear Modeling & Simulation * Nonlinearity in Practice * Nonlinearity in Multi-Physics Systems * Nonlinear Modes and Modal Interactions

This book describes advances in the application of chaos theory to classical scattering and nonequilibrium statistical mechanics generally, and to transport by deterministic diffusion in particular. The author presents the basic tools of dynamical systems theory, such as dynamical instability, topological analysis, periodic-orbit methods, Liouvillian dynamics, dynamical randomness and large-deviation formalism. These tools are applied to chaotic scattering and to transport in systems near equilibrium and maintained out of equilibrium. Chaotic Scattering is illustrated with disk scatterers and with examples of unimolecular chemical reactions and then generalized to transport in spatially extended systems. This book will be bought by researchers interested in chaos, dynamical systems, chaotic scattering, and statistical mechanics in theoretical, computational and mathematical physics and also in theoretical chemistry.

This book deals with methods to evaluate scientific productivity. In the book statistical methods, deterministic and stochastic models and numerous indexes are discussed that will help the reader to understand the nonlinear science dynamics and to be able to develop or construct systems for appropriate evaluation of research productivity and management of research groups and organizations. The dynamics of science structures and systems is complex, and the evaluation of research productivity requires a combination of qualitative and quantitative methods and measures. The book has three parts. The first part is devoted to mathematical models describing the importance of science for economic growth and systems for the evaluation of research organizations of different size. The second part contains descriptions and discussions of numerous indexes for the evaluation of the productivity of researchers and groups of researchers of different size (up to the comparison of research productivities of research communities of nations). Part three contains discussions of non-Gaussian laws connected to scientific productivity and presents various deterministic and stochastic models of science dynamics and research productivity. The book shows that many famous fat tail distributions as well as many deterministic and stochastic models and processes, which are well known from physics, theory of extreme events or population dynamics, occur also in the description of dynamics of scientific systems and in the description of the characteristics of research productivity. This is not a surprise as scientific systems are nonlinear, open and dissipative.

In this monograph, the authors present their recently developed theory of electromagnetic interactions. This neoclassical approach extends the classical electromagnetic theory down to atomic scales and allows the explanation of various non-classical phenomena in the same framework. While the classical Maxwell-Lorentz electromagnetism theory succeeds in describing the physical reality at macroscopic scales, it struggles at atomic scales. Here, quantum mechanics traditionally takes over to describe non-classical phenomena such as the hydrogen spectrum and de Broglie waves. By means of modifying the classical theory, the approach presented here is able to consistently explain quantum-mechanical effects, and while similar to quantum mechanics in some respects, this neoclassical theory also differs markedly from it. In particular, the newly developed framework omits probabilistic interpretations of the wave function and features a new fundamental spatial scale which, at the size of the free electron, is much larger than the classical electron radius and is relevant to plasmonics and emission physics. This book will appeal to researchers interested in advanced aspects of electromagnetic theory. Treating the classical approach in detail, including non-relativistic aspects and the Lagrangian framework, and comparing the neoclassical theory with quantum mechanics and the de Broglie-Bohm theory, this work is completely self-contained.

This prizewinning PhD thesis presents a general discussion of the orbital motion close to solar system small bodies (SSSBs), which induce non-central asymmetric gravitational fields in their neighborhoods. It introduces the methods of qualitative theory in nonlinear dynamics to the study of local/global behaviors around SSSBs. Detailed mechanical models are employed throughout this dissertation, and specific numeric techniques are developed to compensate for the difficulties of directly analyzing. Applying this method, several target systems, like asteroid 216 Kleopatra, are explored in great detail, and the results prove to be both revealing and pervasive for a large group of SSSBs.

This book presents recent research results relating to applications of nonlinear dynamics, focusing specifically on four topics of wide interest: heart dynamics, DNA/RNA, cell mobility, and proteins. The book derives from the First BCAM Workshop on Nonlinear Dynamics in Biological Systems, held in June 2014 at the Basque Center of Applied Mathematics (BCAM). At this international meeting, researchers from different but complementary backgrounds, including molecular dynamics, physical chemistry, bio-informatics and biophysics, presented their most recent results and discussed the future direction of their studies using theoretical, mathematical modeling and experimental approaches. Such was the level of interest stimulated that the decision was taken to produce this publication, with the organizers of the event acting as editors. All of the contributing authors are researchers working on diverse biological problems that can be approached using nonlinear dynamics. The book will appeal especially to applied mathematicians, biophysicists, and computational biologists.

This thesis presents a systematic discussion of experimental approaches to investigating the nonlinear interaction of ultrashort visible strong fields with dielectrics directly in the time domain. The key finding is the distinctly different peak-intensity dependence of the light-matter energy transfer dynamics on the one hand, and the observed transient optical and electronic modifications on the other. As the induced electron dynamics evolve on sub-femtosecond timescales, real-time spectroscopy requires attosecond temporal resolution. This allows a range of parameters to be identified where the optical properties of the samples exposed to ultrashort light fields suffer dramatic changes allowing signal metrology while real absorption leading to dissipation is essentially absent. These findings indicate the feasibility of efficient optical switching at frequencies several orders of magnitude faster than current state-of-the-art electronics and thus have far-reaching technological consequences.