The book deals with the development of continual models of turbulent natural media. Such models serve as a ground for the statement and numerical evaluation of the key problems of the structure and evolution of the numerous astrophysical and geophysical objects. The processes of ordering (self-organization) in an originally chaotic turbulent medium are addressed and treated in detail with the use of irreversible thermodynamics and stochastic dynamics approaches which underlie the respective models. Different examples of ordering set up in the natural environment and outer space are brought and thoroughly discussed, the main focus being given to the protoplanetary discs formation and evolution.

The book contains recent developments and contemporary research in mathematical analysis and in its application to problems arising from the biological and physical sciences. The book is of interest to readers who wish to learn of new research in such topics as linear and nonlinear analysis, mathematical biology and ecology, dynamical systems, graph theory, variational analysis and inequalities, functional analysis, differential and difference equations, partial differential equations, approximation theory, and chaos. All papers were prepared by participants at the International Conference on Recent Advances in Mathematical Biology, Analysis and Applications (ICMBAA-2015) held during 4-6 June 2015 in Aligarh, India. A focal theme of the conference was the application of mathematics to the biological sciences and on current research in areas of theoretical mathematical analysis that can be used as sophisticated tools for the study of scientific problems. The conference provided researchers, academicians and engineers with a platform that encouraged them to exchange their innovative ideas in mathematical analysis and its applications as well as to form interdisciplinary collaborations. The content of the book is divided into three parts: Part I contains contributions from participants whose topics are related to nonlinear dynamics and its applications in biological sciences. Part II has contributions which concern topics on nonlinear analysis and its applications to a variety of problems in science, engineering and industry. Part III consists of contributions dealing with some problems in applied analysis.

This book presents the proceedings of a conference on dynamical systems held in honor of Jurgen Scheurle in January 2012. Through both original research papers and survey articles leading experts in the field offer overviews of the current state of the theory and its applications to mechanics and physics. In particular, the following aspects of the theory of dynamical systems are covered: - Stability and bifurcation - Geometric mechanics and control theory - Invariant manifolds, attractors and chaos - Fluid mechanics and elasticity - Perturbations and multiscale problems - Hamiltonian dynamics and KAM theory Researchers and graduate students in dynamical systems and related fields, including engineering, will benefit from the articles presented in this volume.

All of us are confronted with complex phenomena occurring in daily life and in the living and inanimate nature surrounding us. Our scientific curiosity strives to unravel the mechanisms at work to create such complexity. Among various approaches to solve this problem, the field of synergetics, developed by Hermann Haken, has proven very successful as a general and interdisciplinary concept for describing and explaining complex phenomena that appear in systems under non-equilibrium conditions. These comprise dynamical states in evolving systems, spatial structure-forming processes, synchronization of states and regulatory mechanisms, and many other examples. The encompassing concepts have been applied to many disciplines, like physics, chemistry, biology, and beyond those also from synergetics to information theory, brain science, economics, and others. Starting from basic methods of complexity research and synergetics, this volume contains thirty contributions on complex systems that exhibit spontaneous pattern formation far from thermal equilibrium. Written by international experts and young researchers assembled under one roof, this volume reflects state of the art research from a variety of scientific fields and disciplines where complexity theory and synergetics are important or even indispensable tools today and in the future.

The contributions of this volume stem from the "Fifth International Conference on the Dynamics of Information Systems" held in Gainesville, FL in February 2013, and discuss state-of the-art techniques in handling problems and solutions in the broad field of information systems. Dynamics of Information Systems: Computational and Mathematical Challenges presents diverse aspects of modern information systems with an emphasis on interconnected network systems and related topics, such as signal and message reconstruction, network connectivity, stochastic network analysis, cyber and computer security, community and cohesive structures in complex networks. Information systems are a vital part of modern societies. They are essential to our daily actions, including social networking, business and bank transactions, as well as sensor communications. The rapid increase in these capabilities has enabled us with more powerful systems, readily available to sense, control, disperse, and analyze information.

"Pseudochaotic Kicked Oscillators: Renormalization, Symbolic Dynamics, and Transport" presents recent developments in pseudochaos, which is concerned with complex branching behaviors of dynamical systems at the interface between orderly and chaotic motion. Pseudochaos is characterized by the trapping of orbits in the vicinity of self-similar hierarchies of islands of stability, producing phase-space displacements which increase asymptotically as a power of time. This monograph is a thorough, self-contained investigation of a simple one-dimensional model (a kicked harmonic oscillator) which exhibits pseudochaos in its purest form. It is intended for graduate students and researchers in physics and applied mathematics, as well as specialists in nonlinear dynamics.
Dr. John H. Lowenstein is a Professor Emeritus in the Department of Physics at New York University, USA."

This book contains original articles submitted to the Seventh International Conference on Cognitive Neurodynamics (ICCN 2019). The brain is an endless case study of a complex system characterized by multiple levels of integration, multiple time scales of activity, and multiple coding and decoding properties. The contribution of several disciplines, mathematics, physics, computer science, neurobiology, pharmacology, physiology, and behavioral and clinical sciences, is necessary in order to cope with such seemingly unattainable complexity that transforms the experimental information into a tricky puzzle which hides the correspondence with model predictions. This conference gathered active participants to discuss ideas and pose new questions from different viewpoints, ranging from single neurons and neural networks to animal/human behavior in theoretical and experimental studies. The conference is organized with plenary lectures, mini-symposia, interdisciplinary round tables, and oral and poster sessions.

This book presents the foundation of the theory of almost automorphic functions in abstract spaces and the theory of almost periodic functions in locally and non-locally convex spaces and their applications in differential equations. Since the publication of Almost automorphic and almost periodic functions in abstract spaces (Kluwer Academic/Plenum, 2001), there has been a surge of interest in the theory of almost automorphic functions and applications to evolution equations. Several generalizations have since been introduced in the literature, including the study of almost automorphic sequences, and the interplay between almost periodicity and almost automorphic has been exposed for the first time in light of operator theory, complex variable functions and harmonic analysis methods. As such, the time has come for a second edition to this work, which was one of the most cited books of the year 2001. This new edition clarifies and improves upon earlier materials, includes many relevant contributions and references in new and generalized concepts and methods, and answers the longtime open problem, "What is the number of almost automorphic functions that are not almost periodic in the sense of Bohr?" Open problems in non-locally convex valued almost periodic and almost automorphic functions are also indicated. As in the first edition, materials are presented in a simplified and rigorous way. Each chapter is concluded with bibliographical notes showing the original sources of the results and further reading.

This book provides a comprehensive introduction to the theory of ordinary differential equations with a focus on mechanics and dynamical systems as important applications of the theory. The text is written to be used in the traditional way or in a more applied way. In addition to its use in a traditional one or two semester graduate course in mathematics, the book is organized to be used for interdisciplinary courses in applied mathematics, physics, and engineering.

In this thesis, the author investigates the biophysical basis of the local field potential (LFP) as a way of gaining a better understanding of its underlying physiological mechanisms. The results represent major advances in our understanding and interpretation of LFPs and brain oscillations. They highlight the importance of using suitable experimental and analytical methods to explore the activity of brain circuits and point to the LFP as a useful, but complex variable for this purpose.

T his book presents a t.hooretical framewerk and control methodology for a class of complcx dyna.mical systenis characterized by high state space dimension, multiple inpu t.s anrl out puts. significant nonlinearity, parametric uncertainty and unmodellod dyuarni cs. The book start.s wit.h an inl.rod uct.orv Chapter 1 where the peculiari- ties of control problcrns Ior complex systems are discussed and motivating examples from different fiolds of seience and technology are given. Chapter 2 prcscnts SO Il I(' rcsults of nonlinear control theory which assist in reading subsequent chaptors. The main notions and concepts of stability theory are int roduced. and problems of nonlinear transformation of sys- tem coordinates an' discussod. On this basis, we consider different design techniques and approaches t 0 linearization. stabilization and passification of nonlinear dynamical SySt('IIIS. Chapter 3 gives an cx posit.ion of the Speed-Gradient method and its ap- plications to nonlinear aud adaptive control. Convergence and robustness properties are exam iued. I~ roblcms of rcgulat ion, tracking, partial stabiliza- tion and control of 11amiItonia.n systerns are considered .

Research into social systems is challenging due to their complex nature. Traditional methods of analysis are often difficult to apply effectively as theories evolve over time. This can be due to a lack of appropriate data, or too much uncertainty. It can also be the result of problems which are not yet understood well enough in the general sense so that they can be classified, and an appropriate solution quickly identified. Simulation is one tool that deals well with these challenges, fits in well with the deductive process, and is useful for testing theory. This field is still relatively new, and much of the work is necessarily innovative, although it builds upon a rich and varied foundation. There are a number of existing modelling paradigms being applied to complex social systems research. Additionally, new methods and measures are being devised through the process of conducting research. We expect that readers will enjoy the collection of high quality research works from new and accomplished researchers.

With many areas of science reaching across their boundaries and becoming more and more interdisciplinary, students and researchers in these fields are confronted with techniques and tools not covered by their particular education. Especially in the life- and neurosciences quantitative models based on nonlinear dynamics and complex systems are becoming as frequently implemented as traditional statistical analysis. Unfamiliarity with the terminology and rigorous mathematics may discourage many scientists to adopt these methods for their own work, even though such reluctance in most cases is not justified.

This book bridges this gap by introducing the procedures and methods used for analyzing nonlinear dynamical systems. In Part I, the concepts of fixed points, phase space, stability and transitions, among others, are discussed in great detail and implemented on the basis of example elementary systems. Part II is devoted to specific, non-trivial applications: coordination of human limb movement (Haken-Kelso-Bunz model), self-organization and pattern formation in complex systems (Synergetics), and models of dynamical properties of neurons (Hodgkin-Huxley, Fitzhugh-Nagumo and Hindmarsh-Rose). Part III may serve as a refresher and companion of some mathematical basics that have been forgotten or were not covered in basic math courses. Finally, the appendix contains an explicit derivation and basic numerical methods together with some programming examples as well as solutions to the exercises provided at the end of certain chapters. Throughout this book all derivations are as detailed and explicit as possible, and everybody with some knowledge of calculus should be able to extract meaningful guidance follow and apply the methods of nonlinear dynamics to their own work.

"This book is a masterful treatment, one might even say a gift, to the interdisciplinary scientist of the future."

"With the authoritative voice of a genuine practitioner, Fuchs is a master teacher of how to handle complex dynamical systems."

"What I find beautiful in this book is its clarity, the clear definition of terms, every step explained simply and systematically."

(J.A.Scott Kelso, excerpts from the foreword)

This book presents a proof of universal computation in the Game of Life cellular automaton by using a Turing machine construction. It provides an introduction including background information and an extended review of the literature for Turing Machines, Counter Machines and the relevant patterns in Conway's Game of Life so that the subject matter is accessibly to non specialists. The book contains a description of the author's Turing machine in Conway's Game of Life including an unlimited storage tape provided by growing stack structures and it also presents a fast universal Turing machine designed to allow the working to be demonstrated in a convenient period of time.

This volume collects the edited and reviewed contribution presented in the 7th iTi Conference in Bertinoro, covering fundamental and applied aspects in turbulence. In the spirit of the iTi conference, the volume is produced after the conference so that the authors had the opportunity to incorporate comments and discussions raised during the meeting. In the present book, the contributions have been structured according to the topics: I Theory II Wall bounded flows III Pipe flow IV Modelling V Experiments VII Miscellaneous topics

This collection of original articles and surveys addresses the recent advances in linear and nonlinear aspects of the theory of partial differential equations. The key topics include operators as "sums of squares" of real and complex vector fields, nonlinear evolution equations, local solvability, and hyperbolic questions.

The high reliability required in industrial processes has created the necessity of detecting abnormal conditions, called faults, while processes are operating. The term fault generically refers to any type of process degradation, or degradation in equipment performance because of changes in the process's physical characteristics, process inputs or environmental conditions. This book is about the fundamentals of fault detection and diagnosis in a variety of nonlinear systems which are represented by ordinary differential equations. The fault detection problem is approached from a differential algebraic viewpoint, using residual generators based upon high-gain nonlinear auxiliary systems ('observers'). A prominent role is played by the type of mathematical tools that will be used, requiring knowledge of differential algebra and differential equations. Specific theorems tailored to the needs of the problem-solving procedures are developed and proved. Applications to real-world problems, both with constant and time-varying faults, are made throughout the book and include electromechanical positioning systems, the Continuous Stirred Tank Reactor (CSTR), bioreactor models and belt drive systems, to name but a few.

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 Poincare 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, Wurzburg "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

Our everyday life is in?uenced by many unexpected (dif?cult to predict) events usually referred as a chance. Probably, we all are as we are due to the accumulation point of a multitude of chance events. Gambling games that have been known to human beings nearly from the beginning of our civilization are based on chance events. These chance events have created the dream that everybody can easily become rich. This pursuit made gambling so popular. This book is devoted to the dynamics of the mechanical randomizers and we try to solve the problem why mechanical device (roulette) or a rigid body (a coin or a die) operating in the way described by the laws of classical mechanics can behave in such a way and produce a pseudorandom outcome. During mathematical lessons in primary school we are taught that the outcome of the coin tossing experiment is random and that the probability that the tossed coin lands heads (tails) up is equal to 1/2. Approximately, at the same time during physics lessons we are told that the motion of the rigid body (coin is an example of suchabody)isfullydeterministic. Typically,studentsarenotgiventheanswertothe question Why this duality in the interpretation of the simple mechanical experiment is possible? Trying to answer this question we describe the dynamics of the gambling games based on the coin toss, the throw of the die, and the roulette run.

Inverse scattering theory is a major theme in applied mathematics, with applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The inverse scattering problem is both nonlinear and ill-posed, thus presenting challenges in the development of efficient inversion algorithms. A further complication is that anisotropic materials cannot be uniquely determined from given scattering data. In the first edition of Inverse Scattering Theory and Transmission Eigenvalues, the authors discussed methods for determining the support of inhomogeneous media from measured far field data and the role of transmission eigenvalue problems in the mathematical development of these methods. In this second edition, three new chapters describe recent developments in inverse scattering theory. In particular, the authors explore the use of modified background media in the nondestructive testing of materials and methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data. They also examine nonscattering wave numbers-a subset of transmission eigenvalues-using techniques taken from the theory of free boundary value problems for elliptic partial differential equations and discuss the dualism of scattering poles and transmission eigenvalues that has led to new methods for the numerical computation of scattering poles. This book will be of interest to research mathematicians and engineers and physicists working on problems in target identification. It will also be useful to advanced graduate students in many areas of applied mathematics.

Complexity science has been a source of new insight in physical and social systems and has demonstrated that unpredictability and surprise are fundamental aspects of the world around us. This book is the outcome of a discussion meeting of leading scholars and critical thinkers with expertise in complex systems sciences and leaders from a variety of organizations, sponsored by the Prigogine Center at The University of Texas at Austin and the Plexus Institute, to explore strategies for understanding uncertainty and surprise. Besides contributions to the conference, it includes a key digest by the editors as well as a commentary by the late nobel laureate Ilya Prigogine, "Surprises in half of a century." The book is intended for researchers and scientists in complexity science, as well as for a broad interdisciplinary audience of both practitioners and scholars. It will well serve those interested in the research issues and in the application of complexity science to physical and social systems.

This volume collects the edited and reviewed contribution presented in the 9th iTi Conference that took place virtually, covering fundamental and applied aspects in turbulence. In the spirit of the iTi conference, the volume is produced after the conference so that the authors had the opportunity to incorporate comments and discussions raised during the meeting. In the present book, the contributions have been structured according to the topics: I Experiments II Simulations and Modelling III Data Processing and Scaling IV Theory V Miscellaneous topics

Nonlinear Systems and Methods For Mechanical, Electrical and Biosystems presents topics observed at the 3rd Conference on Nonlinear Science and Complexity(NSC), focusing on energy transfer and synchronization in hybrid nonlinear systems. The studies focus on fundamental theories and principles, analytical and symbolic approaches, computational techniques in nonlinear physical science and mathematics. Broken into three parts, the text covers: Parametrical excited pendulum, nonlinear dynamics in hybrid systems, dynamical system synchronization and (N+1) body dynamics as well as new views different from the existing results in nonlinear dynamics, mathematical methods for dynamical systems including conservation laws, dynamical symmetry in nonlinear differential equations and invex energies and nonlinear phenomena in physical problems such as solutions, complex flows, chemical kinetics, Toda lattices and parallel manipulator.

This book is useful to scholars, researchers and advanced technical members of industrial laboratory facilities developing new tools and products.

The dimension theory of dynamical systems has progressively developed, especially over the last two decades, into an independent and extremely active field of research. Its main aim is to study the complexity of sets and measures that are invariant under the dynamics. In particular, it is essential to characterizing chaotic strange attractors. To date, some parts of the theory have either only been outlined, because they can be reduced to the case of maps, or are too technical for a wider audience. In this respect, the present monograph is intended to provide a comprehensive guide. Moreover, the text is self-contained and with the exception of some basic results in Chapters 3 and 4, all the results in the book include detailed proofs. The book is intended for researchers and graduate students specializing in dynamical systems who wish to have a sufficiently comprehensive view of the theory together with a working knowledge of its main techniques. The discussion of some open problems is also included in the hope that it may lead to further developments. Ideally, readers should have some familiarity with the basic notions and results of ergodic theory and hyperbolic dynamics at the level of an introductory course in the area, though the initial chapters also review all the necessary material.

This monograph develops a generalised energy flow theory to investigate non-linear dynamical systems governed by ordinary differential equations in phase space and often met in various science and engineering fields. Important nonlinear phenomena such as, stabilities, periodical orbits, bifurcations and chaos are tack-led and the corresponding energy flow behaviors are revealed using the proposed energy flow approach. As examples, the common interested nonlinear dynamical systems, such as, Duffing's oscillator, Van der Pol's equation, Lorenz attractor, Roessler one and SD oscillator, etc, are discussed. This monograph lights a new energy flow research direction for nonlinear dynamics. A generalised Matlab code with User Manuel is provided for readers to conduct the energy flow analysis of their nonlinear dynamical systems. Throughout the monograph the author continuously returns to some examples in each chapter to illustrate the applications of the discussed theory and approaches. The book can be used as an undergraduate or graduate textbook or a comprehensive source for scientists, researchers and engineers, providing the statement of the art on energy flow or power flow theory and methods.