This book intends to introduce some recent results on passivity of complex dynamical networks with single weight and multiple weights. The book collects novel research ideas and some definitions in complex dynamical networks, such as passivity, output strict passivity, input strict passivity, finite-time passivity, and multiple weights. Furthermore, the research results previously published in many flagship journals are methodically edited and presented in a unified form. The book is likely to be of interest to university researchers and graduate students in Engineering and Mathematics who wish to study the passivity of complex dynamical networks.

This monograph focuses on the design of personalized and adaptive online interactive learning environment (OILE) to enhance students' learning in and about complex dynamic systems (CDS). Numerous studies show that students experience difficulties when learning in and about CDS. The difficulties are due to challenges originating from a) the structural complexity of CDS, (b) the production of dynamic behavior from the underlying systems structure, and (c) methods, techniques and tools employed in the analysis of such systems. Despite the fact that studies have uncovered such learning challenges, it is still not well understood how we may effectively address these challenges. In this monograph, the authors provide some answers as to how we may best improve our cognitive capabilities to meet these challenges by way of effective instructional methods, techniques, and tools and their implementation in the form of an OILE. The OILE developed for this purpose, builds on a five-step holistic instructional design framework; identification of instructional design models, identification of authentic learning material, identification of instructional methods, identification of instructional techniques, and design of the interface and implementation of the tool. In this OILE development, six well-documented instructional design models were considered; a four component instructional design, first principles of instruction, constructivists learning environment, task centered instruction, cognitive apprenticeship, and elaboration theory.

The second edition of the book includes a new chapter on the study of composition operators on the Hardy space and their complete characterization by Gordon and Hedenmalm. The book is devoted to Diophantine approximation, the analytic theory of Dirichlet series and their composition operators, and connections between these two domains which often occur through the Kronecker approximation theorem and the Bohr lift. The book initially discusses Harmonic analysis, including a sharp form of the uncertainty principle, Ergodic theory and Diophantine approximation, basics on continued fractions expansions, and the mixing property of the Gauss map and goes on to present the general theory of Dirichlet series with classes of examples connected to continued fractions, Bohr lift, sharp forms of the Bohnenblust-Hille theorem, Hardy-Dirichlet spaces, composition operators of the Hardy-Dirichlet space, and much more. Proofs throughout the book mix Hilbertian geometry, complex and harmonic analysis, number theory, and ergodic theory, featuring the richness of analytic theory of Dirichlet series. This self-contained book benefits beginners as well as researchers.

The time variability of many natural and social phenomena is not well described by standard methods of data analysis. However, nonlinear time series analysis uses chaos theory and nonlinear dynamics to understand seemingly unpredictable behavior. The results are applied to real data from physics, biology, medicine, and engineering in this volume. Researchers from all experimental disciplines, including physics, the life sciences, and the economy, will find the work helpful in the analysis of real world systems. First Edition Hb (1997): 0-521-55144-7 First Edition Pb (1997): 0-521-65387-8

This book reports on the latest numerical and experimental findings in the field of high-lift technologies. It covers interdisciplinary research subjects relating to scientific computing, aerodynamics, aeroacoustics, material sciences, aircraft structures, and flight mechanics. The respective chapters are based on papers presented at the Final Symposium of the Collaborative Research Center (CRC) 880, which was held on December 17-18, 2019 in Braunschweig, Germany. The conference and the research presented here were partly supported by the CRC 880 on "Fundamentals of High Lift for Future Civil Aircraft," funded by the DFG (German Research Foundation). The papers offer timely insights into high-lift technologies for short take-off and landing aircraft, with a special focus on aeroacoustics, efficient high-lift, flight dynamics, and aircraft design.

The theory of dynamical systems has given rise to the vast new area variously called applied dynamics, nonlinear science, or chaos theory. This introductory text covers the central topological and probabilistic notions in dynamics ranging from Newtonian mechanics to coding theory. The only prerequisite is a basic undergraduate analysis course. The authors use a progression of examples to present the concepts and tools for describing asymptotic behavior in dynamical systems, gradually increasing the level of complexity. Subjects include contractions, logistic maps, equidistribution, symbolic dynamics, mechanics, hyperbolic dynamics, strange attractors, twist maps, and KAM-theory.

This volume arose from a semester at CIRM-Luminy on "Thermodynamic Formalism: Applications to Probability, Geometry and Fractals" which brought together leading experts in the area to discuss topical problems and recent progress. It includes a number of surveys intended to make the field more accessible to younger mathematicians and scientists wishing to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic theory and dynamical system and its applications to other topics, particularly Riemannian geometry (especially in negative curvature), statistical properties of dynamical systems and fractal geometry. This work will be of value both to graduate students and more senior researchers interested in either learning about the main ideas and themes in thermodynamic formalism, and research themes which are at forefront of research in this area.

Caustics are natural phenomena, forming light patterns in rainbows or through drinking glasses, and creating light networks at the bottom of swimming pools. Only in recent years have scientists started to artificially create simple caustics with laser light. However, these realizations have already contributed to progress in advanced imaging, lithography, and micro-manipulation. In this book, Alessandro Zannotti pioneers caustics in many ways, establishing the field of artificial caustic optics. He employs caustic design to customize high-intensity laser light. This is of great relevance for laser-based machining, sensing, microscopy, and secure communication. The author also solves a long standing problem concerning the origin of rogue waves which appear naturally in the sea and can have disastrous consequences. By means of a far-reaching optical analogy, he identifies scattering of caustics in random media as the origin of rogue waves, and shows how nonlinear light-matter interaction increases their probability.

The proceedings of the 2017 Symposium on Chaos, Complexity and Leadership illuminate current research results and academic work from the fields of physics, mathematics, education, economics, as well as management and social sciences. The text explores chaotic and complex systems, as well as chaos and complexity theory in view of their applicability to management and leadership. This proceedings explores non-linearity as well as data-modelling and simulation in order to uncover new approaches and perspectives. Effort will not be spared in bringing theory into practice while exploring leadership and management-laden concepts. This book will cover the analysis of different chaotic developments from different fields within the concepts of chaos and complexity theory. Researchers and students in the field will find answers to questions surrounding these intertwined and compelling fields.

Topics in Modal Analysis & Testing, Volume 8: Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics, 2020, the eighth volume of nine from the Conference, brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Modal Analysis, including papers on:Operational Modal & Modal Analysis Applications Experimental Techniques Modal Analysis, Measurements & Parameter Estimation Modal Vectors & Modeling Basics of Modal Analysis Additive Manufacturing & Modal Testing of Printed Parts

This collection of peer-reviewed workshop papers provides comprehensive coverage of cutting-edge research into topological approaches to data analysis and visualization. It encompasses the full range of new algorithms and insights, including fast homology computation, comparative analysis of simplification techniques, and key applications in materials and medical science. The book also addresses core research challenges such as the representation of large and complex datasets, and integrating numerical methods with robust combinatorial algorithms. In keeping with the focus of the TopoInVis 2017 Workshop, the contributions reflect the latest advances in finding experimental solutions to open problems in the sector. They provide an essential snapshot of state-of-the-art research, helping researchers to keep abreast of the latest developments and providing a basis for future work. Gathering papers by some of the world's leading experts on topological techniques, the book represents a valuable contribution to a field of growing importance, with applications in disciplines ranging from engineering to medicine.

This book discusses fundamentally new biomedical imaging methods, such as holography, holographic and resonant interferometry, and speckle optics. It focuses on the development of holographic interference microscopy and its use in the study of phase objects such as nerve and muscle fibers subjected to the influence of laser radiation, magnetic fields, and hyperbaric conditions. The book shows how the myelin sheath and even the axon itself exhibit waveguide properties, enabling a fresh new look at the mechanisms of information transmission in the human body. The book presents theoretically and experimentally tested holographic and speckle-optical methods and devices used for investigating complex, diffusely scattering surfaces such as skin and muscle tissue. Additionally, it gives broad discussion of the authors' own original fundamental and applied research dedicated to helping physicians introduce new contact-less methods of diagnosis and treatment of diseases of the cardiovascular and neuromuscular systems into medical practice. The book is aimed at a broad spectrum of scientific specialists in the fields of speckle optics, holography, laser physics, morphology and cytochemistry, as well as medical professionals such as physiologists, neuropathologists, neurosurgeons, cardiologists and dentists.

Nonlinear Structures & Systems, Volume 1: Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics, 2020, the first volume of eight from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Nonlinear Dynamics, including papers on: Nonlinear Reduced-order Modeling Jointed Structures: Identification, Mechanics, Dynamics Experimental Nonlinear Dynamics Nonlinear Model & Modal Interactions Nonlinear Damping Nonlinear Modeling & Simulation Nonlinearity & System Identification

This book contains contributions from the participants of the research group hosted by the ZiF - Center for Interdisciplinary Research at the University of Bielefeld during the period 2013-2017 as well as from the conclusive conference organized at Bielefeld in December 2017. The contributions consist of original research papers: they mirror the scientific developments fostered by this research program or the state-of-the-art results presented during the conclusive conference. The volume covers current research in the areas of operator theory and dynamical systems on networks and their applications, indicating possible future directions. The book will be interesting to researchers focusing on the mathematical theory of networks; it is unique as, for the first time, continuous network models - a subject that has been blooming in the last twenty years - are studied alongside more classical and discrete ones. Thus, instead of two different worlds often growing independently without much intercommunication, a new path is set, breaking with the tradition. The fruitful and beneficial exchange of ideas and results of both communities is reflected in this book.

This book is aimed to make careful analysis to various mathematical problems derived from shock reflection by using the theory of partial differential equations. The occurrence, propagation and reflection of shock waves are important phenomena in fluid dynamics. Comparing the plenty of studies of physical experiments and numerical simulations on this subject, this book makes main efforts to develop the related theory of mathematical analysis, which is rather incomplete so far. The book first introduces some basic knowledge on the system of compressible flow and shock waves, then presents the concept of shock polar and its properties, particularly the properties of the shock polar for potential flow equation, which are first systematically presented and proved in this book. Mathematical analysis of regular reflection and Mach reflection in steady and unsteady flow are the most essential parts of this book. To give challenges in future research, some long-standing open problems are listed in the end. This book is attractive to researchers in the fields of partial differential equations, system of conservation laws, fluid dynamics, and shock theory.

This book provides a broad introduction to the subject of dynamical systems, suitable for a one or two-semester graduate course. In the first chapter, the authors introduce over a dozen examples, and then use these examples throughout the book to motivate and clarify the development of the theory. Topics include topological dynamics, symbolic dynamics, ergodic theory, hyperbolic dynamics, one-dimensional dynamics, complex dynamics, and measure-theoretic entropy. The authors top off the presentation with some beautiful and remarkable applications of dynamical systems to areas such as number theory, data storage, and internet search engines.

This book, now in its second edition, introduces the singularity analysis of differential and difference equations via the Painleve test and shows how Painleve analysis provides a powerful algorithmic approach to building explicit solutions to nonlinear ordinary and partial differential equations. It is illustrated with integrable equations such as the nonlinear Schroedinger equation, the Korteweg-de Vries equation, Henon-Heiles type Hamiltonians, and numerous physically relevant examples such as the Kuramoto-Sivashinsky equation, the Kolmogorov-Petrovski-Piskunov equation, and mainly the cubic and quintic Ginzburg-Landau equations. Extensively revised, updated, and expanded, this new edition includes: recent insights from Nevanlinna theory and analysis on both the cubic and quintic Ginzburg-Landau equations; a close look at physical problems involving the sixth Painleve function; and an overview of new results since the book's original publication with special focus on finite difference equations. The book features tutorials, appendices, and comprehensive references, and will appeal to graduate students and researchers in both mathematics and the physical sciences.

This book provides a complete exposition of equidistribution and counting problems weighted by a potential function of common perpendicular geodesics in negatively curved manifolds and simplicial trees. Avoiding any compactness assumptions, the authors extend the theory of Patterson-Sullivan, Bowen-Margulis and Oh-Shah (skinning) measures to CAT(-1) spaces with potentials. The work presents a proof for the equidistribution of equidistant hypersurfaces to Gibbs measures, and the equidistribution of common perpendicular arcs between, for instance, closed geodesics. Using tools from ergodic theory (including coding by topological Markov shifts, and an appendix by Buzzi that relates weak Gibbs measures and equilibrium states for them), the authors further prove the variational principle and rate of mixing for the geodesic flow on metric and simplicial trees-again without the need for any compactness or torsionfree assumptions. In a series of applications, using the Bruhat-Tits trees over non-Archimedean local fields, the authors subsequently prove further important results: the Mertens formula and the equidistribution of Farey fractions in function fields, the equidistribution of quadratic irrationals over function fields in their completions, and asymptotic counting results of the representations by quadratic norm forms. One of the book's main benefits is that the authors provide explicit error terms throughout. Given its scope, it will be of interest to graduate students and researchers in a wide range of fields, for instance ergodic theory, dynamical systems, geometric group theory, discrete subgroups of locally compact groups, and the arithmetic of function fields.

This book presents exact, closed-form solutions for the response of a variety of nonlinear oscillators (free, damped, forced). The solutions presented are expressed in terms of special functions. To help the reader understand these `non-standard' functions, detailed explanations and rich illustrations of their meanings and contents are provided. In addition, it is shown that these exact solutions in certain cases comprise the well-known approximate solutions for some nonlinear oscillations.

This work provides a convincing motivation for and introduction to magnon-based computing. The challenges faced by the conventional semiconductor-transistor-based computing industry are contrasted with the many exciting avenues for developing spin waves (or magnons) as a complementary technology wherein information can be encoded, transmitted, and operated upon: essential ingredients for any computing paradigm. From this general foundation, one particular operation is examined: phase conjugation via four-wave-mixing (FWM). The author constructs an original theory describing the generation of a phase conjugate mirror with the remarkable property that any incident spin wave will be reflected back along the same direction of travel. After establishing a theoretical framework, the careful design of the experiment is presented, followed by the demonstration of a magnetic phase conjugate mirror using four-wave mixing for the first time. The thesis concludes with an investigation into the unexpected fractal behaviour observed arising from the phase conjugate mirror - a result that is testament to the richness and vibrancy of these highly nonlinear spin wave systems.

The main goal of this book is to systematically address the mathematical methods that are applied in the study of synchronization of infinite-dimensional evolutionary dissipative or partially dissipative systems. It bases its unique monograph presentation on both general and abstract models and covers several important classes of coupled nonlinear deterministic and stochastic PDEs which generate infinite-dimensional dissipative systems. This text, which adapts readily to advanced graduate coursework in dissipative dynamics, requires some background knowledge in evolutionary equations and introductory functional analysis as well as a basic understanding of PDEs and the theory of random processes. Suitable for researchers in synchronization theory, the book is also relevant to physicists and engineers interested in both the mathematical background and the methods for the asymptotic analysis of coupled infinite-dimensional dissipative systems that arise in continuum mechanics.

This book focuses on bifurcation and stability in nonlinear discrete systems, including monotonic and oscillatory stability. It presents the local monotonic and oscillatory stability and bifurcation of period-1 fixed-points on a specific eigenvector direction, and discusses the corresponding higher-order singularity of fixed-points. Further, it explores the global analysis of monotonic and oscillatory stability of fixed-points in 1-dimensional discrete systems through 1-dimensional polynomial discrete systems. Based on the Yin-Yang theory of nonlinear discrete systems, the book also addresses the dynamics of forward and backward nonlinear discrete systems, and the existence conditions of fixed-points in said systems. Lastly, in the context of local analysis, it describes the normal forms of nonlinear discrete systems and infinite-fixed-point discrete systems. Examining nonlinear discrete systems from various perspectives, the book helps readers gain a better understanding of the nonlinear dynamics of such systems.

This book provides a comprehensive overview of statistical descriptions of turbulent flows. Its main objectives are to point out why ordinary perturbative treatments of the Navier-Stokes equation have been rather futile, and to present recent advances in non-perturbative treatments, e.g., the instanton method and a stochastic interpretation of turbulent energy transfer. After a brief introduction to the basic equations of turbulent fluid motion, the book outlines a probabilistic treatment of the Navier-Stokes equation and chiefly focuses on the emergence of a multi-point hierarchy and the notion of the closure problem of turbulence. Furthermore, empirically observed multiscaling features and their impact on possible closure methods are discussed, and each is put into the context of its original field of use, e.g., the renormalization group method is addressed in relation to the theory of critical phenomena. The intended readership consists of physicists and engineers who want to get acquainted with the prevalent concepts and methods in this research area.

This book seeks to bridge the gap between the parlance, the models, and even the notations used by physicists and those used by mathematicians when it comes to the topic of probability and stochastic processes. The opening four chapters elucidate the basic concepts of probability, including probability spaces and measures, random variables, and limit theorems. Here, the focus is mainly on models and ideas rather than the mathematical tools. The discussion of limit theorems serves as a gateway to extensive coverage of the theory of stochastic processes, including, for example, stationarity and ergodicity, Poisson and Wiener processes and their trajectories, other Markov processes, jump-diffusion processes, stochastic calculus, and stochastic differential equations. All these conceptual tools then converge in a dynamical theory of Brownian motion that compares the Einstein-Smoluchowski and Ornstein-Uhlenbeck approaches, highlighting the most important ideas that finally led to a connection between the Schroedinger equation and diffusion processes along the lines of Nelson's stochastic mechanics. A series of appendices cover particular details and calculations, and offer concise treatments of particular thought-provoking topics.

This book discusses basic topics in the spectral theory of dynamical systems. It also includes two advanced theorems, one by H. Helson and W. Parry, and another by B. Host. Moreover, Ornstein's family of mixing rank-one automorphisms is given with construction and proof. Systems of imprimitivity and their relevance to ergodic theory are also examined. Baire category theorems of ergodic theory, scattered in literature, are discussed in a unified way in the book. Riesz products are introduced and applied to describe the spectral types and eigenvalues of rank-one automorphisms. Lastly, the second edition includes a new chapter "Calculus of Generalized Riesz Products", which discusses the recent work connecting generalized Riesz products, Hardy classes, Banach's problem of simple Lebesgue spectrum in ergodic theory and flat polynomials.