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This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-te

Differentiable dynamical systems. --- Global analysis (Mathematics) --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Topological dynamics --- Differentiable dynamical systems

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The theory and applications of infinite dimensional dynamical systems have attracted the attention of scientists for quite some time. Dynamical issues arise in equations that attempt to model phenomena that change with time. The infi­ nite dimensional aspects occur when forces that describe the motion depend on spatial variables, or on the history of the motion. In the case of spatially depen­ dent problems, the model equations are generally partial differential equations, and problems that depend on the past give rise to differential-delay equations. Because the nonlinearities occurring in thse equations need not be small, one needs good dynamical theories to understand the longtime behavior of solutions. Our basic objective in writing this book is to prepare an entree for scholars who are beginning their journey into the world of dynamical systems, especially in infinite dimensional spaces. In order to accomplish this, we start with the key concepts of a semiflow and a flow. As is well known, the basic elements of dynamical systems, such as the theory of attractors and other invariant sets, have their origins here.

Differentiable dynamical systems. --- Evolution equations. --- Differentiable dynamical systems --- Evolution equations --- Mathematical analysis. --- Analysis (Mathematics). --- Topology. --- Statistical physics. --- Dynamical systems. --- Analysis. --- Complex Systems. --- Statistical Physics and Dynamical Systems. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Mathematical statistics --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- 517.1 Mathematical analysis --- Mathematical analysis --- Statistical methods

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This book provides a broad introduction to the subject of dynamical systems, suitable for a one- or two-semester graduate course. In the first chapter, the authors introduce over a dozen examples, and then use these examples throughout the book to motivate and clarify the development of the theory. Topics include topological dynamics, symbolic dynamics, ergodic theory, hyperbolic dynamics, one-dimensional dynamics, complex dynamics, and measure-theoretic entropy. The authors top off the presentation with some beautiful and remarkable applications of dynamical systems to such areas as number theory, data storage, and Internet search engines. This book grew out of lecture notes from the graduate dynamical systems course at the University of Maryland, College Park, and reflects not only the tastes of the authors, but also to some extent the collective opinion of the Dynamics Group at the University of Maryland, which includes experts in virtually every major area of dynamical systems.

Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Analyse globale (mathématiques) --- Systèmes dynamiques. --- Analyse globale (mathématiques) --- Systèmes dynamiques --- Entropie --- Theorie ergodique

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As systems evolve, they are subjected to random operating environments. In addition, random errors occur in measurements of their outputs and in their design and fabrication where tolerances are not precisely met. This book develops methods for describing random dynamical systems, and it illustrates how the methods can be used in a variety of applications. The first half of the book concentrates on finding approximations to random processes using the methodologies of probability theory. The second half of the book derives approximations to solutions of various problems in mechanics, electronic circuits, population biology, and genetics. In each example, the underlying physical or biological phenomenon is described in terms of nonrandom models taken from the literature, and the impact of random noise on the solutions is investigated. The mathematical problems in these applicitons involve random pertubations of gradient systems, Hamiltonian systems, toroidal flows, Markov chains, difference equations, filters, and nonlinear renewal equations. The models are analyzed using the approximation methods described here and are visualized using MATLAB-based computer simulations. This book will appeal to those researchers and graduate students in science and engineering who require tools to investigate stochastic systems.

Perturbation (Mathematics) --- Differentiable dynamical systems. --- Distribution (Probability theory. --- Engineering mathematics. --- Global analysis (Mathematics). --- Mathematics. --- Mechanics, applied. --- Probability Theory and Stochastic Processes. --- Mathematical and Computational Engineering. --- Analysis. --- Applications of Mathematics. --- Theoretical and Applied Mechanics. --- Theoretical, Mathematical and Computational Physics. --- Differentiable dynamical systems --- Probabilities. --- Applied mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Mechanics. --- Mechanics, Applied. --- Mathematical physics. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Perturbation equations --- Perturbation theory --- Approximation theory --- Dynamics --- Functional analysis --- Mathematical physics

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This book provides an approach to the study of perturbation and discretization effects on the long-time behavior of dynamical and control systems. It analyzes the impact of time and space discretizations on asymptotically stable attracting sets, attractors, asumptotically controllable sets and their respective domains of attractions and reachable sets. Combining robust stability concepts from nonlinear control theory, techniques from optimal control and differential games and methods from nonsmooth analysis, both qualitative and quantitative results are obtained and new algorithms are developed, analyzed and illustrated by examples.

Attractors (Mathematics) --- Differentiable dynamical systems. --- Asymptotic expansions. --- Perturbation (Mathematics) --- Differentiable dynamical systems --- Asymptotic expansions --- Geometry --- Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Dynamics. --- Ergodic theory. --- System theory. --- Numerical analysis. --- Calculus of variations. --- Dynamical Systems and Ergodic Theory. --- Systems Theory, Control. --- Numerical Analysis. --- Calculus of Variations and Optimal Control; Optimization. --- Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Philosophy