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In the analysis and synthesis of contemporary systems, engineers and scientists are frequently confronted with increasingly complex models that may simultaneously include components whose states evolve along continuous time and discrete instants; components whose descriptions may exhibit nonlinearities, time lags, transportation delays, hysteresis effects, and uncertainties in parameters; and components that cannot be described by various classical equations, as in the case of discrete-event systems, logic commands, and Petri nets. The qualitative analysis of such systems requires results for finite-dimensional and infinite-dimensional systems; continuous-time and discrete-time systems; continuous continuous-time and discontinuous continuous-time systems; and hybrid systems involving a mixture of continuous and discrete dynamics. Filling a gap in the literature, this textbook presents the first comprehensive stability analysis of all the major types of system models described above. Throughout the book, the applicability of the developed theory is demonstrated by means of many specific examples and applications to important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, artificial neural networks (with and without time delays), digital signal processing, a class of discrete-event systems (with applications to manufacturing and computer load balancing problems) and a multicore nuclear reactor model. The book covers the following four general topics: * Representation and modeling of dynamical systems of the types described above* Presentation of Lyapunov and Lagrange stability theory for dynamical systems defined on general metric spaces* Specialization of this stability theory to finite-dimensional dynamical systems* Specialization of this stability theory to infinite-dimensional dynamical systems Replete with exercises and requiring basic knowledge of linear algebra, analysis, and differential equations, the work may be used as a textbook for graduate courses in stability theory of dynamical systems. The book may also serve as a self-study reference for graduate students, researchers, and practitioners in applied mathematics, engineering, computer science, physics, chemistry, biology, and economics. [Publisher]

Differentiable dynamical systems --- Stability --- Differentiable dynamical systems. --- Stability. --- Dynamique différentiable --- Systèmes dynamiques --- Stabilité --- 517.9 --- Dynamics --- Mechanics --- Motion --- Vibration --- Benjamin-Feir instability --- Equilibrium --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Dynamique différentiable. --- Systèmes dynamiques. --- Stabilité.

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This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs. As is typical in any subject combining Diophantine problems and geometry, a fundamental goal is to describe arithmetic properties, at least qualitatively, in terms of underlying geometric structures. Key features: - Provides an entry for graduate students into an active field of research - Provides a standard reference source for researchers - Includes numerous exercises and examples - Contains a description of many known results and conjectures, as well as an extensive glossary, bibliography, and index This graduate-level text assumes familiarity with basic algebraic number theory. Other topics, such as basic algebraic geometry, elliptic curves, nonarchimedean analysis, and the theory of Diophantine approximation, are introduced and referenced as needed. Mathematicians and graduate students will find this text to be an excellent reference.

Mathematics. --- Dynamical Systems and Ergodic Theory. --- Differentiable dynamical systems. --- Mathématiques --- Dynamique différentiable --- Dynamics --- Number theory --- Applied Mathematics --- Calculus --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Number study --- Numbers, Theory of --- Dynamical systems --- Kinetics --- Data structures (Computer science). --- Dynamics. --- Ergodic theory. --- Data Structures. --- Algebra --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Data structures (Computer scienc. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Data structures (Computer science) --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- Electronic data processing --- File organization (Computer science) --- Abstract data types (Computer science) --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems. --- Artificial intelligence --- Dynamical Systems. --- Data Science. --- Data processing.

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Topological bifurcation theory is one of the most essential topics in mathematics. This book contains original bifurcation results for the existence of oscillations and chaotic behaviour of differential equations and discrete dynamical systems under variation of involved parameters. Using topological degree theory and a perturbation approach in dynamical systems, a broad variety of nonlinear problems are studied, including: non-smooth mechanical systems with dry frictions; weakly coupled oscillators; systems with relay hysteresis; differential equations on infinite lattices of Frenkel-Kontorova and discretized Klein-Gordon types; blue sky catastrophes for reversible dynamical systems; buckling of beams; and discontinuous wave equations. Precise and complete proofs, together with concrete applications with many stimulating and illustrating examples, make this book valuable to both the applied sciences and mathematical fields, ensuring the book should not only be of interest to mathematicians but to physicists and theoretically inclined engineers interested in bifurcation theory and its applications to dynamical systems and nonlinear analysis.

Mathematics. --- Analysis. --- Topology. --- Dynamical Systems and Ergodic Theory. --- Mechanics. --- Vibration, Dynamical Systems, Control. --- Global analysis (Mathematics). --- Differentiable dynamical systems. --- Vibration. --- Mathématiques --- Analyse globale (Mathématiques) --- Dynamique différentiable --- Topologie --- Mécanique --- Vibration --- Bifurcation theory. --- Bifurcation theory --- Topology --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Mathematical analysis. --- Analysis (Mathematics). --- Dynamics. --- Ergodic theory. --- Dynamical systems. --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Classical Mechanics. --- Cycles --- Mechanics --- Sound --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Statics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- 517.1 Mathematical analysis --- Mathematical analysis --- Calculus. --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Functions --- Geometry, Infinitesimal

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From July 11th to July 22nd, 2005, a NATO advanced study institute, as part of the series “Seminaire ´ de mathematiques ´ superieures”, ´ was held at the U- versite ´ de Montreal, ´ on the subject Equidistribution in the theory of numbers. There were about one hundred participants from sixteen countries around the world. This volume presents details of the lecture series that were given at the school. Across the broad panorama of topics that constitute modern number t- ory one nds shifts of attention and focus as more is understood and better questions are formulated. Over the last decade or so we have noticed incre- ing interest being paid to distribution problems, whether of rational points, of zeros of zeta functions, of eigenvalues, etc. Although these problems have been motivated from very di?erent perspectives, one nds that there is much in common, and presumably it is healthy to try to view such questions as part of a bigger subject. It is for this reason we decided to hold a school on “Equidistribution in number theory” to introduce junior researchers to these beautiful questions, and to determine whether di?erent approaches can in uence one another. There are far more good problems than we had time for in our schedule. We thus decided to focus on topics that are clearly inter-related or do not requirealotofbackgroundtounderstand.

Geometry --- Harmonic analysis. Fourier analysis --- informatietheorie --- Mathematical physics --- Number theory --- Ergodic theory. Information theory --- differentiaalvergelijkingen --- landmeetkunde --- Fourieranalyse --- getallenleer --- Irregularities of distribution (Number theory) --- Irrégularités de distribution (Théorie des nombres) --- Congresses --- Congrès --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B --- Congresses. --- Number theory. --- Geometry, algebraic. --- Differentiable dynamical systems. --- Mathematics. --- Fourier analysis. --- Number Theory. --- Algebraic Geometry. --- Dynamical Systems and Ergodic Theory. --- Measure and Integration. --- Fourier Analysis. --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Algebraic geometry --- Number study --- Numbers, Theory of --- Algebra --- Algebraic geometry. --- Dynamics. --- Ergodic theory. --- Measure theory. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Ergodic transformations --- Continuous groups --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Equidistribution

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This volume—dedicated to Michael K. Sain on the occasion of his seventieth birthday—is a collection of chapters covering recent advances in stochastic optimal control theory and algebraic systems theory. Written by experts in their respective fields, the chapters are thematically organized into four parts: * Part I focuses on statistical control theory, where the cost function is viewed as a random variable and performance is shaped through cost cumulants. In this respect, statistical control generalizes linear-quadratic-Gaussian and H-infinity control. * Part II addresses algebraic systems theory, reviewing the use of algebraic systems over semirings, modules of zeros for linear multivariable systems, and zeros in linear time-delay systems. * Part III discusses advances in dynamical systems characteristics. The chapters focus on the stability of a discontinuous dynamical system, approximate decentralized fixed modes, direct optimal adaptive control, and stability of nonlinear systems with limited information. * Part IV covers engineering education and includes a unique chapter on theology and engineering, one of Sain's latest research interests. The book will be a useful reference for researchers and graduate students in systems and control, algebraic systems theory, and applied mathematics. Requiring only knowledge of undergraduate-level control and systems theory, the work may be used as a supplementary textbook in a graduate course on optimal control or algebraic systems theory.

Mathematics. --- Systems Theory, Control. --- Control, Robotics, Mechatronics. --- Calculus of Variations and Optimal Control; Optimization. --- Applications of Mathematics. --- Game Theory, Economics, Social and Behav. Sciences. --- Dynamical Systems and Ergodic Theory. --- Differentiable dynamical systems. --- Systems theory. --- Mathematical optimization. --- Mathématiques --- Dynamique différentiable --- Optimisation mathématique --- Nonlinear control theory. --- Stochastic control theory. --- System analysis. --- Stochastic control theory --- Nonlinear control theory --- System analysis --- Differentiable dynamical systems --- Civil & Environmental Engineering --- Operations Research --- Engineering & Applied Sciences --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Network theory --- Systems analysis --- Algebra. --- Applied mathematics. --- Engineering mathematics. --- System theory. --- Calculus of variations. --- Probabilities. --- Control engineering. --- Robotics. --- Mechatronics. --- Probability Theory and Stochastic Processes. --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Network analysis --- Network science --- System theory --- Mathematical optimization --- Control theory --- Nonlinear theories --- Stochastic processes --- Distribution (Probability theory. --- Systems, Theory of --- Systems science --- Science --- Math --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- Mathematics --- Philosophy --- Mechanical engineering --- Microelectronics --- Microelectromechanical systems --- Automation --- Machine theory --- Engineering --- Engineering analysis --- Control engineering --- Control equipment --- Engineering instruments --- Programmable controllers --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Isoperimetrical problems --- Variations, Calculus of