The contributions in this volume aim to deepen understanding of some of the current research problems and theories in modern topics such as calculus of variations, optimization theory, complex analysis, real analysis, differential equations, and geometry. Applications to these areas of mathematics are presented within the broad spectrum of research in Engineering Science with particular emphasis on equilibrium problems, complexity in numerical optimization, dynamical systems, non-smooth optimization, complex network analysis, statistical models and data mining, and energy systems. Additional emphasis is given to interdisciplinary research, although subjects are treated in a unified and self-contained manner. The presentation of methods, theory and applications makes this tribute an invaluable reference for teachers, researchers, and other professionals interested in pure and applied research, philosophy of mathematics, and mathematics education. Some review papers published in this volume will be particularly useful for a broader audience of readers as well as for graduate students who search for the latest information. Constantin Caratheodory's wide-ranging influence in the international mathematical community was seen during the first Fields Medals awards at the International Congress of Mathematicians, Oslo, 1936. Two medals were awarded, one to Lars V. Ahlfors and one to Jesse Douglass. It was Caratheodory who presented both their works during the opening of the International Congress. This volume contains significant papers in Science and Engineering dedicated to the memory of Constantin Caratheodory and the spirit of his mathematical influence.

This is the first book focusing on bifurcation dynamics in 1-dimensional polynomial nonlinear discrete systems. It comprehensively discusses the general mathematical conditions of bifurcations in polynomial nonlinear discrete systems, as well as appearing and switching bifurcations for simple and higher-order singularity period-1 fixed-points in the 1-dimensional polynomial discrete systems. Further, it analyzes the bifurcation trees of period-1 to chaos generated by period-doubling, and monotonic saddle-node bifurcations. Lastly, the book presents methods for period-2 and period-doubling renormalization for polynomial discrete systems, and describes the appearing mechanism and period-doublization of period-n fixed-points on bifurcation trees for the first time, offering readers fascinating insights into recent research results in nonlinear discrete systems.

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 text is comprised of selected research articles developed from a workshop on Ergodic Theory, Probabilistic Methods and Applications, held in April 2012 at the Banff International Research Station. It contains contributions from world leading experts in ergodic theory, numerical dynamical systems, molecular dynamics and ocean/atmosphere dynamics, nonequilibrium statistical mechanics. The volume will serve as a valuable reference for mathematicians, physicists, engineers, biologists and climate scientists, who currently use, or wish to learn how to use, probabilistic techniques to cope with dynamical models that display open or non-equilibrium behaviour.

This book, the first dedicated to the topic, provides a comprehensive treatment of forward stimulated Brillouin scattering (SBS) in standard optical fibers. SBS interactions between guided light and sound waves have drawn much attention for over fifty years, and optical fibers provide an excellent playground for the study of Brillouin scattering as they support guided modes of both wave types and provide long interaction lengths. This book is dedicated to forward SBS processes that are driven by co-propagating optical fields. The physics of forward SBS is explained in detail, starting from the fundamentals of interactions between guided optical and acoustic waves, with emphasis given to the acoustic modes that are stimulated in the processes. The realization of forward SBS in standard single-mode, polarization-maintaining and multi-core fibers is then discussed in depth. Innovative potential applications in sensors, monitoring of coating layers, lasers, and radio-frequency oscillators are presented. This book introduces the subject to graduate students in optics and applied physics, and it will be of interest to scientists working in fiber-optics, nonlinear optics and opto-mechanics. Provides the first treatment of forward stimulated Brillouin scattering (SBS) in book form; Reflects the dramatic recent increase in interest in forward SBS processes , driven in part by the promise of new fiber sensing concepts; Delivers a solid and comprehensive grounding in the physics of forward SBS along with detailed experimental set-ups, measurement protocols, and applications.

The study of hyperbolic systems is a core theme of modern dynamics. On surfaces the theory of the ?ne scale structure of hyperbolic invariant sets and their measures can be described in a very complete and elegant way, and is the subject of this book, largely self-contained, rigorously and clearly written. It covers the most important aspects of the subject and is based on several scienti?c works of the leading research workers in this ?eld. This book ?lls a gap in the literature of dynamics. We highly recommend it for any Ph.D student interested in this area. The authors are well-known experts in smooth dynamical systems and ergodic theory. Now we give a more detailed description of the contents: Chapter1.TheIntroductionisadescriptionofthemainconceptsinhyp- bolic dynamics that are used throughout the book. These are due to Bowen, Hirsch, Man' "e, Palis, Pugh, Ruelle, Shub, Sinai, Smale and others. Stable and r unstable manifolds are shown to beC foliated. This result is very useful in a number of contexts. The existence of smooth orthogonal charts is also proved. This chapter includes proofs of extensions to hyperbolic di?eomorphisms of some results of Man' "e for Anosov maps. Chapter 2. All the smooth conjugacy classes of a given topological model are classi?ed using Pinto's and Rand's HR structures. The a?ne structures of Ghys and Sullivan on stable and unstable leaves of Anosov di?eomorphisms are generalized.

This book provides an introduction to dynamical systems with multiple time scales. The approach it takes is to provide an overview of key areas, particularly topics that are less available in the introductory form. The broad range of topics included makes it accessible for students and researchers new to the field to gain a quick and thorough overview. The first of its kind, this book merges a wide variety of different mathematical techniques into a more unified framework. The book is highly illustrated with many examples and exercises and an extensive bibliography. The target audience of this book are senior undergraduates, graduate students as well as researchers interested in using the multiple time scale dynamics theory in nonlinear science, either from a theoretical or a mathematical modeling perspective.

This work introduces a new method for analysing measured signals: nonlinear mode decomposition, or NMD. It justifies NMD mathematically, demonstrates it in several applications and explains in detail how to use it in practice. Scientists often need to be able to analyse time series data that include a complex combination of oscillatory modes of differing origin, usually contaminated by random fluctuations or noise. Furthermore, the basic oscillation frequencies of the modes may vary in time; for example, human blood flow manifests at least six characteristic frequencies, all of which wander in time. NMD allows us to separate these components from each other and from the noise, with immediate potential applications in diagnosis and prognosis. Mat Lab codes for rapid implementation are available from the author. NMD will most likely come to be used in a broad range of applications.

This volume highlights contributions of women mathematicians in the study of complex materials and includes both original research papers and reviews. The featured topics and methods draw on the fields of Calculus of Variations, Partial Differential Equations, Functional Analysis, Differential Geometry and Topology, as well as Numerical Analysis and Mathematical Modelling. Areas of applications include foams, fluid-solid interactions, liquid crystals, shape-memory alloys, magnetic suspensions, failure in solids, plasticity, viscoelasticity, homogenization, crystallization, grain growth, and phase-field models.

At the end of the nineteenth century Lyapunov and Poincare developed the so called qualitative theory of differential equations and introduced geometric- topological considerations which have led to the concept of dynamical systems. In its present abstract form this concept goes back to G.D. Birkhoff. This is also the starting point of Chapter 1 of this book in which uncontrolled and controlled time-continuous and time-discrete systems are investigated. Controlled dynamical systems could be considered as dynamical systems in the strong sense, if the controls were incorporated into the state space. We, however, adapt the conventional treatment of controlled systems as in control theory. We are mainly interested in the question of controllability of dynamical systems into equilibrium states. In the non-autonomous time-discrete case we also consider the problem of stabilization. We conclude with chaotic behavior of autonomous time discrete systems and actual real-world applications.

This book marks the 60th birthday of Prof. Vladimir Erofeev - a well-known specialist in the field of wave processes in solids, fluids, and structures. Featuring a collection of papers related to Prof. Erofeev's contributions in the field, it presents articles on the current problems concerning the theory of nonlinear wave processes in generalized continua and structures. It also discusses a number of applications as well as various discrete and continuous dynamic models of structures and media and problems of nonlinear acoustic diagnostics.

This book shows that evolutionary game theory can unravel how mutual cooperation, trust, and credit in a group emerge in organizations and institutions. Some organizations and institutions, such as insurance unions, credit unions, and banks, originated from very simple mutual-aid groups. Members in these early-stage mutual-aid groups help each other, making rules to promote cooperation, and suppressing free riders. Then, they come to "trust" not only each other but also the group they belong to, itself. The division of labor occurs when the society comes to have diversity and complexity in a larger group, and the division of labor also requires mutual cooperation and trust among different social roles. In a larger group, people cannot directly interact with each other, and the reputation of unknown people helps other decide who is a trustworthy person. However, if gossip spreads untruths about a reputation, trust and cooperation are destroyed. Therefore, how to suppress untrue gossip is also important for trust and cooperation in a larger group. If trustworthiness and credibility can be established, these groups are successfully sustainable. Some develop and evolve and then mature into larger organizations and institutions. Finally, these organizations and institutions become what they are now. Therefore, not only cooperation but also trust and credit are keys to understanding these organizations and institutions. The evolution of cooperation, a topic of research in evolutionary ecology and evolutionary game theory, can be applied to understanding how to make institutions and organizations sustainable, trustworthy, and credible. It provides us with the idea that evolutionary game theory is a good mathematical tool to analyze trust and credit. This kind of research can be applied to current hot topics such as microfinance and the sustainable use of ecosystems.

This book, based on a selection of invited presentations from a topical workshop, focusses on time-variable oscillations and their interactions. The problem is challenging, because the origin of the time variability is usually unknown. In mathematical terms, the oscillations are non-autonomous, reflecting the physics of open systems where the function of each oscillator is affected by its environment. Time-frequency analysis being essential, recent advances in this area, including wavelet phase coherence analysis and nonlinear mode decomposition, are discussed. Some applications to biology and physiology are described. Although the most important manifestation of time-variable oscillations is arguably in biology, they also crop up in, e.g. astrophysics, or for electrons on superfluid helium. The book brings together the research of the best international experts in seemingly very different disciplinary areas.

Chaos and nonlinear dynamics initially developed as a new emergent field with its foundation in physics and applied mathematics. The highly generic, interdisciplinary quality of the insights gained in the last few decades has spawned myriad applications in almost all branches of science and technology-and even well beyond. Wherever quantitative modeling and analysis of complex, nonlinear phenomena is required, chaos theory and its methods can play a key role. This third volume concentrates on reviewing further relevant contemporary applications of chaotic nonlinear systems as they apply to the various cutting-edge branches of engineering. This encompasses, but is not limited to, topics such fluctuation relations and chaotic dynamics in physics, fractals and their applications in epileptic seizures, as well as chaos synchronization. Featuring contributions from active and leading research groups, this collection is ideal both as a reference and as a 'recipe book' full of tried and tested, successful engineering applications.

Stochastic elasticity is a fast developing field that combines nonlinear elasticity and stochastic theories in order to significantly improve model predictions by accounting for uncertainties in the mechanical responses of materials. However, in contrast to the tremendous development of computational methods for large-scale problems, which have been proposed and implemented extensively in recent years, at the fundamental level, there is very little understanding of the uncertainties in the behaviour of elastic materials under large strains. Based on the idea that every large-scale problem starts as a small-scale data problem, this book combines fundamental aspects of finite (large-strain) elasticity and probability theories, which are prerequisites for the quantification of uncertainties in the elastic responses of soft materials. The problems treated in this book are drawn from the analytical continuum mechanics literature and incorporate random variables as basic concepts along with mechanical stresses and strains. Such problems are interesting in their own right but they are also meant to inspire further thinking about how stochastic extensions can be formulated before they can be applied to more complex physical systems.

The subject of these two volumes is non-linear filtering (prediction and smoothing) theory and its application to the problem of optimal estimation, control with incomplete data, information theory, and sequential testing of hypothesis. The book is not only addressed to mathematicians but should also serve the interests of other scientists who apply probabilistic and statistical methods in their work. The theory of martingales presented in the book has an independent interest in connection with problems from financial mathematics. In the second edition, the authors have made numerous corrections, updating every chapter, adding two new subsections devoted to the Kalman filter under wrong initial conditions, as well as a new chapter devoted to asymptotically optimal filtering under diffusion approximation. Moreover, in each chapter a comment is added about the progress of recent years.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics," "CFD," "completely integrable systems," "chaos, synergetics and large-scale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

This book focuses on the nonlinear dynamics based on the vector fields with univariate quadratic functions. This book is a unique monograph for two-dimensional quadratic nonlinear systems. It provides different points of view about nonlinear dynamics and bifurcations of the quadratic dynamical systems. Such a two-dimensional dynamical system is one of simplest dynamical systems in nonlinear dynamics, but the local and global structures of equilibriums and flows in such two-dimensional quadratic systems help us understand other nonlinear dynamical systems, which is also a crucial step toward solving the Hilbert's sixteenth problem. Possible singular dynamics of the two-dimensional quadratic systems are discussed in detail. The dynamics of equilibriums and one-dimensional flows in two-dimensional systems are presented. Saddle-sink and saddle-source bifurcations are discussed, and saddle-center bifurcations are presented. The infinite-equilibrium states are switching bifurcations for nonlinear systems. From the first integral manifolds, the saddle-center networks are developed, and the networks of saddles, source, and sink are also presented. This book serves as a reference book on dynamical systems and control for researchers, students, and engineering in mathematics, mechanical, and electrical engineering.

This the second volume of five from the 28th IMAC on Structural Dynamics and Renewable Energy, 2010, bringing together 17 chapters on Applications of Non-Linear Dynamics. It presents early findings from experimental and computational investigations on Non-Linear Dynamics including studies on Dynamics of a System of Coupled Oscillators with Geometrically Nonlinear Damping, Assigning the Nonlinear Distortions of a Two-input Single-output System, A Multi-harmonic Approach to Updating Locally Nonlinear Structures, A Block Rocking on a Seesawing Foundation, and Enhanced Order Reduction of Forced Nonlinear Systems Using New Ritz Vectors.

The book is dedicated to the construction of particular solutions of systems of ordinary differential equations in the form of series that are analogous to those used in Lyapunov s first method. A prominent place is given to asymptotic solutions that tend to an equilibrium position, especially in the strongly nonlinear case, where the existence of such solutions can t be inferred on the basis of the first approximation alone.

The book is illustrated with a large number of concrete examples of systems in which the presence of a particular solution of a certain class is related to special properties of the system s dynamic behavior. It is a book for students and specialists who work with dynamical systems in the fields of mechanics, mathematics, and theoretical physics.

The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Editor-in-Chief Jurgen Appell, Wurzburg, Germany Honorary and Advisory Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Alfonso Vignoli, Rome, Italy Editorial Board Manuel del Pino, Bath, UK, and Santiago, Chile Mikio Kato, Nagano, Japan Wojciech Kryszewski, Torun, Poland Vicentiu D. Radulescu, Krakow, Poland Simeon Reich, Haifa, Israel Please submit book proposals to Jurgen Appell. Titles in planning include Lucio Damascelli and Filomena Pacella, Morse Index of Solutions of Nonlinear Elliptic Equations (2019) Tomasz W. Dlotko and Yejuan Wang, Critical Parabolic-Type Problems (2019) Rafael Ortega, Periodic Differential Equations in the Plane: A Topological Perspective (2019) Ireneo Peral Alonso and Fernando Soria, Elliptic and Parabolic Equations Involving the Hardy-Leray Potential (2020) Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020) Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)

This book provides an introduction to classical celestial mechanics. It is based on lectures delivered by the authors over many years at both Padua University (MC) and V.N. Karazin Kharkiv National University (EB). The book aims to provide a mathematical description of the gravitational interaction of celestial bodies. The approach to the problem is purely formal. It allows the authors to write equations of motion and solve them to the greatest degree possible, either exactly or by approximate techniques, when there is no other way. The results obtained provide predictions that can be compared with the observations. Five chapters are supplemented by appendices that review certain mathematical tools, deepen some questions (so as not to interrupt the logic of the mainframe with heavy technicalities), give some examples, and provide an overview of special functions useful here, as well as in many other fields of physics. The authors also present the original investigation of torus potential. This book is aimed at senior undergraduate students of physics or astrophysics, as well as graduate students undertaking a master's degree or Ph.D.

This book starts with an introduction to the basic concepts of multistability, then illustrates how multistability arises in different systems and explains the main mechanisms of multistability emergence. A special attention is given to noise which can convert a multistable deterministic system to a monostable stochastic one. Furthermore, the most important applications of multistability in different areas of science, engineering and technology are given attention throughout the book, including electronic circuits, lasers, secure communication, and human perception. The book aims to provide a first approach to multistability for readers, who are interested in understanding its fundamental concepts and applications in several fields. This book will be useful not only to researchers and engineers focusing on interdisciplinary studies, but also to graduate students and technicians. Both theoreticians and experimentalists will rely on it, in fields ranging from mathematics and laser physics to neuroscience and astronomy. The book is intended to fill a gap in the literature, to stimulate new discussions and bring some fundamental issues to a deeper level of understanding of the mechanisms underlying self-organization of matter and world complexity.

The book presents nonlinear, chaotic and fractional dynamics, complex systems and networks, together with cutting-edge research on related topics. The fifteen chapters - written by leading scientists working in the areas of nonlinear, chaotic, and fractional dynamics, as well as complex systems and networks - offer an extensive overview of cutting-edge research on a range of topics, including fundamental and applied research. These include but are not limited to, aspects of synchronization in complex dynamical systems, universality features in systems with specific fractional dynamics, and chaotic scattering. As such, the book provides an excellent and timely snapshot of the current state of research, blending the insights and experiences of many prominent researchers.

This volume, which coincides with the centennial anniversary of the publication of the celebrated monograph "The General Problem of the Stability Motion" by A.M. Liapunov, reviews the current state of the art of the theory and applications of the Liapunov methods. The text contains an introduction and four chapters. Chapter 2 presents some general results in stability theory. The remaining chapters deal with applications in power engineering, chemical engineering, and in non-engineering fields such as economics and in the modelling of interacting species. The diversity of mathematical tools employed, and the described approach to mathematical modelling provide considerations for applications in many other fields. The text is suitable for mathematicians and engineers whose work involves the study and applications of stability theory in systems.