The asymptotic behaviour, in particular "stability" in some sense, is studied systematically for discrete and for continuous linear dynamical systems on Banach spaces. Of particular concern is convergence to an equilibrium with respect to various topologies. Parallels and differences between the discrete and the continuous situation are emphasised.

The aim of this work is to bridge the gap between the well-known Newtonian mechanics and the studies on chaos, ordinarily reserved to experts. Several topics are treated: Lagrangian, Hamiltonian and Jacobi formalisms, studies of integrable and quasi-integrable systems. The chapter devoted to chaos also enables a simple presentation of the KAM theorem. All the important notions are recalled in summaries of the lectures. They are illustrated by many original problems, stemming from real-life situations, the solutions of which are worked out in great detail for the benefit of the reader.

This book will be of interest to undergraduate students as well as others whose work involves mechanics, physics and engineering in general.

This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a "backstepping" method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control. Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

After a short introduction to the fundamentals, this book provides a detailed account of major advances in applying fractional calculus to dynamical systems. Fractional order dynamical systems currently continue to gain further importance in many areas of science and engineering. As with many other approaches to mathematical modeling, the first issue to be addressed is the need to couple a definition of the fractional differentiation or integration operator with the types of dynamical systems that are analyzed. As such, for the fundamentals the focus is on basic aspects of fractional calculus, in particular stability analysis, which is required to tackle synchronization in coupled fractional order systems, to understand the essence of estimators for related integer order systems, and to keep track of the interplay between synchronization and parameter observation. This serves as the common basis for the more advanced topics and applications presented in the subsequent chapters, which include an introduction to the 'Immersion and Invariance' (I&I) methodology, the masterslave synchronization scheme for partially known nonlinear fractional order systems, Fractional Algebraic Observability (FAO) and Fractional Generalized quasi-Synchronization (FGqS) to name but a few. This book is intended not only for applied mathematicians and theoretical physicists, but also for anyone in applied science dealing with complex nonlinear systems.

The aim of this Book is to give an overview, based on the results of nearly three decades of intensive research, of transient chaos. One belief that motivates us to write this book is that, transient chaos may not have been appreciated even within the nonlinear-science community, let alone other scientific disciplines.

This book deals with the application of modern control theory to some important underactuated mechanical systems. It presents modelling and control of the following systems:- the inverted pendulum- a convey-crane system- the pendubot system- the Furuta pendulum- the inertia wheel pendulum- the planar flexible-joint robot- the planar manipulator with two prismatic and one revolute joints- the ball & beam system- the hovercraft model- the planar vertical and take-off landing (PVTOL) aircraft- the helicopter model on a platform- the helicopter modelIn every case the model is obtained in detail using either the Euler-Lagrange formulation or the Newton's second law. We develop control algorithms for every particular system using techniques such as passivity, energy-based Lyapunov functions, forwarding, backstepping or feedback linearization techniques. This book will be of great value for PhD students and researchers in the areas of non-linear control systems, mechanical systems, robotics and control of helicopters. It will help the reader gain experience in the modelling of mechanical systems and familiarize with new control methods for non-linear systems.

The subject of the book is to present the modeling, parameter estimation and other aspects of the identification of nonlinear dynamic systems. The treatment is restricted to the input-output modeling approach. Because of the widespread usage of digital computers discrete time methods are preferred. Time domain parameter estimation methods are dealt with in detail, frequency domain and power spectrum procedures are described shortly. The theory is presented from the engineering point of view, and a large number of examples of case studies on the modeling and identifications of real processes illustrate the methods. Almost all processes are nonlinear if they are considered not merely in a small vicinity of the working point. To exploit industrial equipment as much as possible, mathematical models are needed which describe the global nonlinear behavior of the process. If the process is unknown, or if the describing equations are too complex, the structure and the parameters can be determined experimentally, which is the task of identification. The book is divided into seven chapters dealing with the following topics: 1. Nonlinear dynamic process models 2. Test signals for identification 3. Parameter estimation methods 4. Nonlinearity test methods 5. Structure identification 6. Model validity tests 7. Case studies on identification of real processes Chapter I summarizes the different model descriptions of nonlinear dynamical systems.

This book presents advanced case studies that address a range of important issues arising in space engineering. An overview of challenging operational scenarios is presented, with an in-depth exposition of related mathematical modeling, algorithmic and numerical solution aspects. The model development and optimization approaches discussed in the book can be extended also towards other application areas. The topics discussed illustrate current research trends and challenges in space engineering as summarized by the following list: * Next Generation Gravity Missions * Continuous-Thrust Trajectories by Evolutionary Neurocontrol * Nonparametric Importance Sampling for Launcher Stage Fallout * Dynamic System Control Dispatch * Optimal Launch Date of Interplanetary Missions * Optimal Topological Design * Evidence-Based Robust Optimization * Interplanetary Trajectory Design by Machine Learning * Real-Time Optimal Control * Optimal Finite Thrust Orbital Transfers * Planning and Scheduling of Multiple Satellite Missions * Trajectory Performance Analysis * Ascent Trajectory and Guidance Optimization * Small Satellite Attitude Determination and Control * Optimized Packings in Space Engineering * Time-Optimal Transfers of All-Electric GEO Satellites Researchers working on space engineering applications will find this work a valuable, practical source of information. Academics, graduate and post-graduate students working in aerospace, engineering, applied mathematics, operations research, and optimal control will find useful information regarding model development and solution techniques, in conjunction with real-world applications.

The Navier-Stokes equations describe the motion of fluids and are an invaluable addition to the toolbox of every physicist, applied mathematician, and engineer. The equations arise from applying Newton's laws of motion to a moving fluid and are considered, when used in combination with mass and energy conservation rules, to be the fundamental governing equations of fluid motion. They are relevant across many disciplines, from astrophysics and oceanic sciences to aerospace engineering and materials science. This Student's Guide provides a clear and focused presentation of the derivation, significance and applications of the Navier-Stokes equations, along with the associated continuity and energy equations. Designed as a useful supplementary resource for undergraduate and graduate students, each chapter concludes with a selection of exercises intended to reinforce and extend important concepts. Video podcasts demonstrating the solutions in full are provided online, along with written solutions and other additional resources.

Contents:
1. Elliptic Equations 1.1 Introduction 1.2 Monotone Iterates: A Preview 1.3 Monotone Iterative Technique 1.4 Generalized Quasilinearization 1.5 Weakly Coupled Mixed Monotone Systems 1.6 Elliptic Systems in Unbounded Domains 1.7 MIT Systems in Unbounded Domains 1.8 Notes and Comments 2. Parabolic Equations 2.1 Introduction 2.2 Comparision Theorems 2.3 Monotone Iterative Technique 2.4 Generalized Quasilinearization 2.5 Monotone Flows and Mixed Monotone Systems 2.6 GCR for Weakly Coupled Systems 2.7 Stability and Vector Lyapunov Functions 2.8 Notes and Comments 3. Impulsive Parabolic Equations 3.1 Introduction 3.2 Comparison Results for IPS 3.3 Coupled Lower and Upper Solutions 3.4 Generalized Quasilinearization 3.5 Population Dynamics with Impulses 3.6 Notes and Comments 4. Hyperbolic Equations 4.1 Introduction 4.2 VP and Comparison Results 4.3 Monotone Iterative Technique 4.4 The Method of Generalized Quasilinearization 4.5 Notes and Comments 5. Elliptic Equations 5.1 Introduction 5.2 Comparison Result 5.3 MIT: Semilinear Problems 5.4 MIT: Quasilinear Problems 5.5 MIT: Degenerate Problems 5.6 GQ: Semilinear Problems 5.7 GQ: Quasilinear Problem 5.8 GQ: Degenerate Problems 5.9 Notes and Comments 6. Parabolic Equations 6.1 Introduction 6.2 Monotone Iterative Technique 6.3 Generalized Quasilinearization 6.4 Nonlocal Problems 6.5 GQ: Nonlocal Problems 6.6 Quasilinear Problems 6.7 GQ: Quasilinear Problems 6.8 Notes and Comments 7. Hyperbolic Equations 7.1 Introduction 7.2 Notation and Comparison Results 7.3 Monotone Iterative Technique 7.4 Generalized Quasilinearization 7.5 Notes and Comments Appendicies

This volume presents a systematic and unified treatment of Leray-Schauder continuation theorems in nonlinear analysis. In particular, fixed point theory is established for many classes of maps, such as contractive, non-expansive, accretive, and compact maps, to name but a few. This book also presents coincidence and multiplicity results. Many applications of current interest in the theory of nonlinear differential equations are presented to complement the theory. The text is essentially self-contained, so it may also be used as an introduction to topological methods in nonlinear analysis. This volume will appeal to graduate students and researchers in mathematical analysis and its applications.

A century ago, Lewis Fry Richardson introduced the concept of energy cascades in turbulence. Since this conceptual breakthrough, turbulence has been studied in diverse systems and our knowledge has increased considerably through theoretical, numerical, experimental and observational advances. Eddy turbulence and wave turbulence are the two regimes we can find in nature. So far, most attention has been devoted to the former regime, eddy turbulence, which is often observed in water. However, physicists are often interested in systems for which wave turbulence is relevant. This textbook deals with wave turbulence and systems composed of a sea of weak waves interacting non-linearly. After a general introduction which includes a brief history of the field, the theory of wave turbulence is introduced rigorously for surface waves. The theory is then applied to examples in hydrodynamics, plasma physics, astrophysics and cosmology, giving the reader a modern and interdisciplinary view of the subject.

This important new book sets forth a comprehensive description of various mathematical aspects of problems originating in numerical solution of hyperbolic systems of partial differential equations. The authors present the material in the context of the important mechanical applications of such systems, including the Euler equations of gas dynamics, magnetohydrodynamics (MHD), shallow water, and solid dynamics equations. This treatment provides-for the first time in book form-a collection of recipes for applying higher-order non-oscillatory shock-capturing schemes to MHD modelling of physical phenomena.

The authors also address a number of original "nonclassical" problems, such as shock wave propagation in rods and composite materials, ionization fronts in plasma, and electromagnetic shock waves in magnets. They show that if a small-scale, higher-order mathematical model results in oscillations of the discontinuity structure, the variety of admissible discontinuities can exhibit disperse behavior, including some with additional boundary conditions that do not follow from the hyperbolic conservation laws. Nonclassical problems are accompanied by a multiple nonuniqueness of solutions. The authors formulate several selection rules, which in some cases easily allow a correct, physically realizable choice.

This work systematizes methods for overcoming the difficulties inherent in the solution of hyperbolic systems. Its unique focus on applications, both traditional and new, makes Mathematical Aspects of Numerical Solution of Hyperbolic Systems particularly valuable not only to those interested the development of numerical methods, but to physicists and engineers who strive to solve increasingly complicated nonlinear equations.

Nonlinearity plays a major role in the understanding of most physical, chemical, biological, and engineering sciences.
Nonlinear problems fascinate scientists and engineers, but often elude exact treatment. However elusive they may be, the solutions do exist-if only one perseveres in seeking them out.

Self-Similarity and Beyond presents a myriad of approaches to finding exact solutions for a diversity of nonlinear problems. These include group-theoretic methods, the direct method of Clarkson and Kruskal, traveling waves, hodograph methods, balancing arguments, embedding special solutions into a more general class, and the infinite series approach.

The author's approach is entirely constructive. Numerical solutions either motivate the analysis or confirm it, therefore they are treated alongside the analysis whenever possible. Many examples drawn from real physical situations-primarily fluid mechanics and nonlinear diffusion-illustrate and emphasize the central points presented.

Accessible to a broad base of readers, Self-Similarity and Beyond illuminates a variety of productive methods for meeting the challenges of nonlinearity. Researchers and graduate students in nonlinearity, partial differential equations, and fluid mechanics, along with mathematical physicists and numerical analysts, will re-discover the importance of exact solutions and find valuable additions to their mathematical toolkits.

The book is intended for all those who are interested in application problems related to dynamical systems. It provides an overview of recent findings on dynamical systems in the broadest sense. Divided into 46 contributed chapters, it addresses a diverse range of problems. The issues discussed include: Finite Element Analysis of optomechatronic choppers with rotational shafts; computational based constrained dynamics generation for a model of a crane with compliant support; model of a kinetic energy recuperation system for city buses; energy accumulation in mechanical resonance; hysteretic properties of shell dampers; modeling a water hammer with quasi-steady and unsteady friction in viscoelastic conduits; application of time-frequency methods for the assessment of gas metal arc welding conditions; non-linear modeling of the human body's dynamic load; experimental evaluation of mathematical and artificial neural network modeling for energy storage systems; interaction of bridge cables and wake in vortex-induced vibrations; and the Sommerfeld effect in a single DOF spring-mass-damper system with non-ideal excitation.

Written for graduate students and researchers, Nanoscale Hydrodynamics of Simple Systems covers fundamental aspects of nanoscale hydrodynamics and extends this basis to examples. Covering classical, generalised and extended hydrodynamic theories, the title also discusses their limitations. It introduces the reader to nanoscale fluid phenomena and explores how fluid dynamics on this extreme length scale can be understood using hydrodynamic theory and detailed atomistic simulations. It also comes with additional resources including a series of explanatory videos on the installation of the code package, as well as discussion, analysis and visualisations of simulations. This title primarily focusses on training the reader to identify when classical theory breaks down, how to extend and generalise the theory, as well as assimilate how simulations and theory together can be used to gain fundamental knowledge about the fluid dynamics of small-scale systems.

This book is based on proceedings from a February 2004 Santa Fe Institute workshop. Its contributing chapter authors treat the ecology of predator-prey interactions and food web theory, structure, and dynamics, joining researchers who also work on complex systems and on large nonlinear networks, from the points of view of other sub-fields within ecology. Food webs play a central role in the debates on the role of complexity in stability, persistence, and resilience. Better empirical data and the exploding interest in the subject of networks across social, physical, and natural sciences prompted creation of this volume. The book explores the boundaries of what is known of the relationship between structure and dynamics in ecological networks, and defines directions for future developments in this field.

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.

Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel's 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits.

This book is aimed at mathematicians, scientists, and engineers, studying models that involve a discontinuity, or studying the theory of nonsmooth systems for its own sake. It is divided in two complementary courses: piecewise smooth flows and maps, respectively. Starting from well known theoretical results, the authors bring the reader into the latest challenges in the field, going through stability analysis, bifurcation, singularities, decomposition theorems and an introduction to kneading theory. Both courses contain many examples which illustrate the theoretical concepts that are introduced.

Cellular automata were introduced in the first half of the last century by John von Neumann who used them as theoretical models for self-reproducing machines. The authors present a self-contained exposition of the theory of cellular automata on groups and explore its deep connections with recent developments in geometric group theory, symbolic dynamics, and other branches of mathematics and theoretical computer science. The topics treated include in particular the Garden of Eden theorem for amenable groups, and the Gromov-Weiss surjunctivity theorem as well as the solution of the Kaplansky conjecture on the stable finiteness of group rings for sofic groups. The volume is entirely self-contained, with 10 appendices and more than 300 exercises, and appeals to a large audience including specialists as well as newcomers in the field. It provides a comprehensive account of recent progress in the theory of cellular automata based on the interplay between amenability, geometric and combinatorial group theory, symbolic dynamics and the algebraic theory of group rings which are treated here for the first time in book form.

This book is an introduction to nonlinear programming. It deals with the theoretical foundations and solution methods, beginning with the classical procedures and reaching up to "modern" methods like trust region methods or procedures for nonlinear and global optimization. A comprehensive bibliography including diverse web sites with information about nonlinear programming, in particular software, is presented. Without sacrificing the necessary mathematical rigor, excessive formalisms are avoided. Several examples, exercises with detailed solutions, and applications are provided, making the text adequate for individual studies. The book is written for students from the fields of applied mathematics, engineering, economy, and computation.

Since its first appearance as a set of lecture notes published by the Courant Institute in 1974, this book served as an introduction to various subjects in nonlinear functional analysis. The current edition is a reprint of these notes, with added bibliographic references. Topological and analytic methods are developed for treating nonlinear ordinary and partial differential equations. The first two chapters of the book introduce the notion of topological degree and develop its basic properties. These properties are used in later chapters in the discussion of bifurcation theory (the possible branching of solutions as parameters vary), including the proof of Rabinowitz's global bifurcation theorem. Stability of the branches is also studied.The book concludes with a presentation of some generalized implicit function theorems of Nash-Moser type with applications to Kolmogorov-Arnold-Moser theory and to conjugacy problems. After more than 20 years, this book continues to be an excellent graduate level textbook and a useful supplementary course text.

This book celebrates the 50th anniversary of the Institute of Mathematics, Statistics and Scientific Computing (IMECC) of the University of Campinas, Brazil, by offering reviews of selected research developed at one of the most prestigious mathematics institutes in Latin America. Written by senior professors at the IMECC, it covers topics in pure and applied mathematics and statistics ranging from differential geometry, dynamical systems, Lie groups, and partial differential equations to computational optimization, mathematical physics, stochastic process, time series, and more. A report on the challenges and opportunities of research in applied mathematics - a highly active field of research in the country - and highlights of the Institute since its foundation in 1968 completes this historical volume, which is unveiled in the same year that the International Mathematical Union (IMU) names Brazil as a member of the Group V of countries with the most relevant contributions in mathematics.

This is an introduction to the dynamics of fluids at small scales, the physical and mathematical underpinnings of Brownian motion, and the application of these subjects to the dynamics and flow of complex fluids such as colloidal suspensions and polymer solutions. It brings together continuum mechanics, statistical mechanics, polymer and colloid science, and various branches of applied mathematics, in a self-contained and integrated treatment that provides a foundation for understanding complex fluids, with a strong emphasis on fluid dynamics. Students and researchers will find that this book is extensively cross-referenced to illustrate connections between different aspects of the field. Its focus on fundamental principles and theoretical approaches provides the necessary groundwork for research in the dynamics of flowing complex fluids.