Reduced order models, or model reduction, have been used in many technologically advanced areas to ensure the associated complicated mathematical models remain computable. For instance, reduced order models are used to simulate weather forecast models and in the design of very large scale integrated circuits and networked dynamical systems. For linear systems, the model reduction problem has been addressed from several perspectives and a comprehensive theory exists. Although many results and efforts have been made, at present there is no complete theory of model reduction for nonlinear systems or, at least, not as complete as the theory developed for linear systems. This monograph presents, in a uniform and complete fashion, moment matching techniques for nonlinear systems. This includes extensive sections on nonlinear time-delay systems; moment matching from input/output data and the limitations of the characterization of moment based on a signal generator described by differential equations. Each section is enriched with examples and is concluded with extensive bibliographical notes. This monograph provides a comprehensive and accessible introduction into model reduction for researchers and students working on non-linear systems.

This book provides research summaries from a number of different focuses in Mathematics, and compiles biographical sketches of top professionals in this important field.

This book provides new research on the advances in non-linear dynamics. Chapter One studies compactions in carbon nanotube arrays. Chapter Two reviews the elastic and plastic type behaviours in the Fractal Theory of Motion at nanoscale. Chapter Three analyses a particular model of tumour progression, assuming that the invasive cells, the connective tissue and the proteases are moving through a non-differential medium governed by the Non-Standard Scale Relativity Theory (NSRT) (Scale Relativity Theory with arbitrary constant fractal dimension). Chapter Four studies the process of drug release from a polymer matrix. Chapter Five examines the implications of drug release from a polymeric matrix process. Chapter Six reviews behaviours of travelling waves and Shapiro step types in a tumour-growth model. Chapter Seven discusses the astonishing evolutionary dynamics of a class of nonlinear discrete 2D pattern formations and growth models.

This book presents a collection of problems for nonlinear dynamics, chaos theory and fractals. Besides the solved problems, supplementary problems are also added. Each chapter contains an introduction with suitable definitions and explanations to tackle the problems.The material is self-contained, and the topics range in difficulty from elementary to advanced. While students can learn important principles and strategies required for problem solving, lecturers will also find this text useful, either as a supplement or text, since concepts and techniques are developed in the problems.

H Robust design is an advancing technology which aims to achieve the system design purpose under intrinsic random fluctuation and external disturbance. This book introduces several robust design methods, some of which include linear to nonlinear systems and frequency to time domain. This book provides not only a complete theoretical development and application of H robust design over the last three decades, but also an integrated platform for control, signal processing, communication, systems and synthetic biology. Based on the theoretical H robust design results, the authors also give some practical design examples to illustrate the procedure and validate the performance of the proposed H method with computational simulations and tables.

Sliding mode control was first introduced in the 1950s. It is a nonlinear control technique with many unique properties. In this book, different aspects of SMC are explored. Chapters include new developments in research on a sliding mode governor for hydropower plants; integral sliding mode control (I-SMC) for a variable speed wind turbine system and a I-SMC method for load frequency control (LFC) of nonlinear power systems with wind turbines; the control of a stand-alone photovoltaic (PV) system; leader-follower-based formation control of a group of mobile robots; the application of Takagi-Sugeno (T-S) fuzzy model in coordinated control of multiple robots system; an induction motor speed control using the nonsingular terminal sliding-mode control method; adaptive nonsingular terminal sliding mode (NTSM) tracking control scheme based on backstepping design presented for Micro-Electro-Mechanical Systems (MEMS) vibratory gryoscopes; and a hybrid actuator and its control using a cascade sliding mode technique.

This book focuses on recent advances made in the field of non-linear dynamic modelling in economics. Mathematically, linearity is a very special kind of relation between variables chosen to the purpose of simplification. Even in physics linear modelling of dynamical systems was a first choice for quite some time, due to convenience in analysis, as exemplified by the acceleration law, the harmonic oscillator, the wave equation, and the like. The methods of analysis were simply developed to the ease of dealing with such systems. These methods are found under the heading of infinitesimal calculus; at early stages dynamical processes were formulated as differential equations in continuous time.

Bose-Einstein condensation was discovered in atomic gas systems, where Bose condensate occupies 100% of the total system at zero temperature. Liquid helium systems have been investigated based on the Landau theory, where the superfluid component of liquid helium is background flow. According to the Landau theory, it is doubtful that the superfluid component is a Bose condensate. In experiments, the probability of helium atoms with zero momentum is a few percent of the total liquid helium at ultra-low temperatures. However, the superfluid component occupies 100% of the liquid helium at zero temperature, as macroscopic observations indicate. This new book introduces a quasi-particle representing an eigenstate of the total Hamiltonian

The book is designed to serve as a textbook for courses offered to upper-undergraduate students enrolled in physics. The first edition of this book was published in 2014. As there is a demand for the next edition, it is quite natural to take note of the several advances that have occurred in the subject over the past five years and to decide which of these are appropriate for inclusion at the textbook level, given the fundamental nature and the significance of the subject area. This is the prime motivation for bringing out a revised second edition. Among the newer mechanisms and materials, the book introduces the super-continuum generation, which arises from an excellent interplay of the various mechanisms of optical nonlinearity. The topics covered in this book are quantum mechanics of nonlinear interaction of matter and radiation, formalism and phenomenology of nonlinear wave mixing processes, optical phase conjugation and applications, self-focusing and self-phase modulation and their role in pulse modification, nonlinear absorption mechanisms, and optical limiting applications, photonic switching and bi-stability, and physical mechanisms leading to a nonlinear response in a variety of materials. This book has emerged from an attempt to address the requirement of presenting the subject at the college level. This textbook includes rigorous features such as the elucidation of relevant basic principles of physics; a clear exposition of the ideas involved at an appropriate level; coverage of the physical mechanisms of non-linearity; updates on physical mechanisms and emerging photonic materials and emphasis on the experimental study of nonlinear interactions. The detailed coverage and pedagogical tools make this an ideal textbook for students and researchers enrolled in physics and related courses.

Just 23 years ago Benoit Mandelbrot published his famous picture of the Mandelbrot set, but that picture has changed our view of the mathematical and physical universe. In this text, Mandelbrot offers 25 papers from the past 25 years, many related to the famous inkblot figure. Of historical interest are some early images of this fractal object produced with a crude dot-matrix printer. The text includes some items not previously published.

The systole of a compact metric space $X$ is a metric invariant of $X$, defined as the least length of a noncontractible loop in $X$. When $X$ is a graph, the invariant is usually referred to as the girth, ever since the 1947 article by W. Tutte. The first nontrivial results for systoles of surfaces are the two classical inequalities of C. Loewner and P. Pu, relying on integral-geometric identities, in the case of the two-dimensional torus and real projective plane, respectively. Currently, systolic geometry is a rapidly developing field, which studies systolic invariants in their relation to other geometric invariants of a manifold. This book presents the systolic geometry of manifolds and polyhedra, starting with the two classical inequalities, and then proceeding to recent results, including a proof of M. Gromov's filling area conjecture in a hyperelliptic setting. It then presents Gromov's inequalities and their generalisations, as well as asymptotic phenomena for systoles of surfaces of large genus, revealing a link both to ergodic theory and to properties of congruence subgroups of arithmetic groups. The author includes results on the systolic manifestations of Massey products, as well as of the classical Lusternik-Schnirelmann category.

The ""infinite-dimensional groups"" in the title refer to unitary groups of Hilbert spaces, the infinite symmetric group, groups of homeomorphisms of manifolds, groups of transformations of measure spaces, etc. The book presents an approach to the study of such groups based on ideas from geometric functional analysis and from exploring the interplay between dynamical properties of those groups, combinatorial Ramsey-type theorems, and the phenomenon of concentration of measure. The dynamics of infinite-dimensional groups is very much unlike that of locally compact groups. For instance, every locally compact group acts freely on a suitable compact space (Veech). By contrast, a 1983 result by Gromov and Milman states that whenever the unitary group of a separable Hilbert space continuously acts on a compact space, it has a common fixed point. In the book, this new fast-growing theory is built strictly from well-understood examples up. The book has no close counterpart and is based on recent research articles. At the same time, it is organized so as to be reasonably self-contained. The topic is essentially interdisciplinary and will be of interest to mathematicians working in geometric functional analysis, topological and ergodic dynamics, Ramsey theory, logic and descriptive set theory, representation theory, topological groups, and operator algebras.

The papers in this volume reflect a broad spectrum of current research activities on the theory and applications of nonlinear dynamics and evolution equations. They are based on lectures given during the International Conference on Nonlinear Dynamics and Evolution Equations at Memorial University of Newfoundland, St. John's, NL, Canada, July 6-10, 2004. This volume contains thirteen invited and refereed papers. Nine of these are survey papers, introducing the reader to, and describing the current state of the art in major areas of dynamical systems, ordinary, functional and partial differential equations, and applications of such equations in the mathematical modelling of various biological and physical phenomena. These papers are complemented by four research papers that examine particular problems in the theory and applications of dynamical systems.

Ergodic theory studies measure-preserving transformations of measure spaces. These objects are intrinsically infinite, and the notion of an individual point or of an orbit makes no sense. Still there are a variety of situations when a measure-preserving transformation (and its asymptotic behavior) can be well described as a limit of certain finite objects (periodic processes). The first part of this book develops this idea systematically. Genericity of approximation in various categories is explored, and numerous applications are presented, including spectral multiplicity and properties of the maximal spectral type.The second part of the book contains a treatment of various constructions of cohomological nature with an emphasis on obtaining interesting asymptotic behavior from approximate pictures at different time scales. The book presents a view of ergodic theory not found in other expository sources. It is suitable for graduate students familiar with measure theory and basic functional analysis.

This book collects select papers presented at the International Workshop and Conference on Topology & Applications, held in Kochi, India, from 9-11 December 2018. The book discusses topics on topological dynamical systems and topological data analysis. Topics are ranging from general topology, algebraic topology, differential topology, fuzzy topology, topological dynamical systems, topological groups, linear dynamics, dynamics of operator network topology, iterated function systems and applications of topology. All contributing authors are eminent academicians, scientists, researchers and scholars in their respective fields, hailing from around the world. The book is a valuable resource for researchers, scientists and engineers from both academia and industry.

/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price

In one of the papers in this collection, the remark that "nothing at all takes place in the universe in which some rule of maximum of minimum does not appear" is attributed to no less an authority than Euler. Simplifying the syntax a little, we might paraphrase this as "Everything is an optimization problem." While this might be something of an overstatement, the element of exaggeration is certainly reduced if we consider the extended form: "Everything is an optimization problem or a system of equations." This observation, even if only partly true, stands as a fitting testimonial to the importance of the work covered by this volume.
Since the 1960s, much effort has gone into the development and application of numerical algorithms for solving problems in the two areas of optimization and systems of equations. As a result, many different ideas have been proposed for dealing efficiently with (for example) severe nonlinearities and/or very large numbers of variables. Libraries of powerful software now embody the most successful of these ideas, and one objective of this volume is to assist potential users in choosing appropriate software for the problems they need to solve. More generally, however, these collected review articles are intended to provide both researchers and practitioners with snapshots of the 'state-of-the-art' with regard to algorithms for particular classes of problem. These snapshots are meant to have the virtues of immediacy through the inclusion of very recent ideas, but they also have sufficient depth of field to show how ideas have developed and how today's research questions have grown out of previous solution attempts.
The most efficient methods for "local optimization, " both unconstrained and constrained, are still derived from the classical Newton approach.
As well as dealing in depth with the various classical, or neo-classical, approaches, the selection of papers on optimization in this volume ensures that newer ideas are also well represented.
Solving nonlinear algebraic systems of equations is closely related to optimization. The two are not completely equivalent, however, and usually something is lost in the translation.
Algorithms for nonlinear equations can be roughly classified as "locally convergent" or "globally convergent." The characterization is not perfect.
Locally convergent algorithms include Newton's method, modern quasi-Newton variants of Newton's method, and trust region methods. All of these approaches are well represented in this volume.

One of the major problems in the study of evolution equations of mathematical physics is the investigation of the behavior of the solutions to these equations when time is large or tends to infinity. The related important questions concern the stability of solutions or the character of the instability if a solution is unstable. In the last few decades, considerable progress in this area has been achieved in the study of autonomous evolution partial differential equations. For a number of basic evolution equations of mathematical physics, it was shown that the long time behavior of their solutions can be characterized by a very important notion of a global attractor of the equation. In this book, the authors study new problems related to the theory of infinite-dimensional dynamical systems that were intensively developed during the last 20 years.They construct the attractors and study their properties for various non-autonomous equations of mathematical physics: the 2D and 3D Navier-Stokes systems, reaction-diffusion systems, dissipative wave equations, the complex Ginzburg-Landau equation, and others. Since, as it is shown, the attractors usually have infinite dimension, the research is focused on the Kolmogorov $\varepsilon$-entropy of attractors. Upper estimates for the $\varepsilon$-entropy of uniform attractors of non-autonomous equations in terms of $\varepsilon$-entropy of time-dependent coefficients are proved. Also, the authors construct attractors for those equations of mathematical physics for which the solution of the corresponding Cauchy problem is not unique or the uniqueness is not proved. The theory of the trajectory attractors for these equations is developed, which is later used to construct global attractors for equations without uniqueness.The method of trajectory attractors is applied to the study of finite-dimensional approximations of attractors. The perturbation theory for trajectory and global attractors is developed and used in the study of the attractors of equations with terms rapidly oscillating with respect to spatial and time variables. It is shown that the attractors of these equations are contained in a thin neighborhood of the attractor of the averaged equation. The book gives systematic treatment to the theory of attractors of autonomous and non-autonomous evolution equations of mathematical physics. It can be used both by specialists and by those who want to get acquainted with this rapidly growing and important area of mathematics.

This is a comprehensive introduction to the exciting scientific field of nonlinear dynamics for students, scientists, and engineers, and requires only minimal prerequisites in physics and mathematics. The book treats all the important areas in the field and provides an extensive and up-to-date bibliography of applications in all fields of science, social science, economics, and even the arts.

This volume presents lectures delivered at a workshop held at the Chinese Academy of Sciences (Bejing). The following articles are included: Nonlinear Systems Control by R. Brockett, Adaptive Control of Discrete-Time Nonlinear Systems with Structural Uncertainties by L.-L. Xie and L. Guo, Networks and Learning by P.R. Kumar, Mathematical Aspects of the Power Control Problem in Mobile Communication Systems by C.W. Sung and W.S. Wong, and Brockett's Problem on Nonlinear Filtering Theory by S.S.-T. Yau. Basic concepts and current research are both presented in the book. The volume offers a comprehensive and easy-to-follow account of many fundamental issues in this diverse field. It should be a suitable text for a graduate course on wireless communication. interested in operations research and mathematical programming.

This volume resulted from a year-long programme at the Morningside Center of Mathematics and the Academia Sinica in Beijing. It presents an overview of nonlinear conversation laws and introduces developments in this expanding field. Zhouping Xin's introductory overview of the subject is followed by lecture notes of leading experts who have made fundamental contributions to this field of research. A. Bressan's theory of $-well-posedness for entropy weak solutions to systems of nonlinear hyperbolic conversation laws in the class of viscosity solutions is one of the most important results in the past two decades; G. Chen discusses weak convergence methods and various applications to many problems; P. Degond details mathematical modelling of semi-conductor devices; B. Perthame describes the theory of asymptotic equivalence between conservation laws and singular kinetic equations; Z. Xin outlines the recent development of the vanishing viscosity problem and nonlinear stability of elementary wave-a major focus of research in the last decade; and the volume concludes with Y. Zheng's lecture on incompressible fluid dynamics.

The contributions to this volume attempt to apply different aspects of Ilya Prigogine's Nobel-prize-winning work on dissipative structures to nonchemical systems as a way of linking the natural and social sciences. They address both the mathematical methods for description of pattern and form as they evolve in biological systems and the mechanisms of the evolution of social systems, containing many variables responding to subjective, qualitative stimuli. The mathematical modeling of human systems, especially those far from thermodynamic equilibrium, must involve both chance and determinism, aspects both quantitative and qualitative. Such systems (and the physical states of matter which they resemble) are referred to as self-organized or dissipative structures in order to emphasize their dependence on the flows of matter and energy to and from their surroundings. Some such systems evolve along lines of inevitable change, but there occur instances of choice, or bifurcation, when chance is an important factor in the qualitative modification of structure. Such systems suggest that evolution is not a system moving toward equilibrium but instead is one which most aptly evokes the patterns of the living world. The volume is truly interdisciplinary and should appeal to researchers in both the physical and social sciences. Based on a workshop on dissipative structures held in 1978 at the University of Texas, contributors include Prigogine, A. G. Wilson, Andre de Palma, D. Kahn, J. L. Deneubourgh, J. W. Stucki, Richard N. Adams, and Erick Jantsch. The papers presented include Allen, "Self-Organization in the Urban System"; Robert Herman, "Remarks on Traffic Flow Theories and the Characterization of Traffic in Cities"; W. H. Zurek and Schieve, "Nucleation Paradigm: Survival Threshold in Population Dynamics"; De Palma et al., "Boolean Equations with Temporal Delays"; Nicholas Georgescu-Roegin, "Energy Analysis and Technology Assessment"; Magoroh Maruyama, "Four Different Causal Meta-types in Biological and Social Sciences"; and Jantsch, "From Self-Reference to Self-Transcendence: The Evolution of Self-Organization Dynamics."

This 4-volume compendium contains the verbatim hard copies of all color slides from the Chua Lecture Series presented at HP in Palo Alto, during the period from September 22 to November 24, 2015. Each lecture consists of 90 minutes, divided into a formal lecture, a discussion session, and an Encore of special trivia that the audience found mesmerizing.These lectures share some unique features of the classic Feynman Lectures on Physics, as much of the materials are presented in the unique style of the author, and the content is original as discovered or invented by the author himself. Unlike most technical books that suffer a notoriously short life span as their features could be superseded by superior models, this series of Chua lectures are intended to never be obsolete - many concepts and principles introduced are in fact new laws of nature, written in the language of sophomore-level mathematics, providing the foundation and the elan vital for initiating and nurturing future concepts and inventions.Volume I - covers everything that a researcher may want to know about memristors but is too afraid to ask.Volume II - shows that memristors can be either volatile or non-volatile, and effectively proving that synapses are non-volatile memristors, while action potentials are generated by locally-active memristors.Volume III - presents an overview of the fascinating phenomenon called chaos, while immersing the audience with the sights and sound of chaos from the Chua Circuit, invented in 1984 by Leon Chua, and has now become the standard textbook example of chaos exhibited by a real nonlinear electronic circuit, and not by computer simulations.Volume IV - surprises the audience with a new law of nature - dubbed the local activity principle, as discovered and proved mathematically in 1996 by Leon Chua. In particular, a Corollary of Chua's local activity theorem, dubbed the edge of chaos, is shown via insightful examples to be the originator of most complex phenomena, including intelligence, creativity, and deep learning. The edge of chaos is Mother Nature's tool for overcoming the tyranny of the second law of thermodynamics by providing an escape hatch for entropy to decrease over time. Indeed, the local activity principle which is profusely illustrated in the final volume, is widely recognized as a new law of thermodynamics, and is identified as the sine qua non of all complex phenomena, including life itself.Exclusive Access to the accompanying Video and Audio materials comes with the purchase of this book.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python's extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students' programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

This thesis deals with specific features of the theory of holomorphic dynamics in dimension 2 and then sets out to study analogous questions in higher dimensions, e.g. dealing with normal forms for rigid germs, and examples of Kato 3-folds. The local dynamics of holomorphic maps around critical points is still not completely understood, in dimension 2 or higher, due to the richness of the geometry of the critical set for all iterates. In dimension 2, the study of the dynamics induced on a suitable functional space (the valuative tree) allows a classification of such maps up to birational conjugacy, reducing the problem to the special class of rigid germs, where the geometry of the critical set is simple. In some cases, from such dynamical data one can construct special compact complex surfaces, called Kato surfaces, related to some conjectures in complex geometry.