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In this book, methodology of dynamical systems theory is applied to investigate the physics of the large-scale ocean circulation. Topics include the dynamics of western boundary currents such as the Gulf Stream in the Atlantic Ocean and the Kurosio in the Pacific Ocean, the stability of the thermohaline circulation, and the El Niño/Southern Oscillation phenomenon in the Tropical Pacific. The book also deals with the numerical methods to apply bifurcation analysis on large-dimensional dynamical systems, with tens of thousands (or more) degrees of freedom, which arise through discretization of ocean and climate models. The novel approach to understand the phenomena of climate variability is through a systematic analysis of the solution structure of a hierarchy of models using these techniques. In this way, a connection between the results of the different models within the hierarchy can be established. Mechanistic description of the physics of the results is provided and, where possible, links with results of state-of-the-art ocean (and climate) models and observations are sought. The reader is expected to have a background in basic fluid dynamics and applied mathematics, although the level of the text sometimes is quite introductory. Each of the chapters is rather self-contained and many details of derivations are provided. Exercises presented at the end of each chapter make it a perfect graduate-level text. This book is aimed at graduate students and researchers in meteorology, oceanography and related fields who are interested in tackling fundamental problems in dynamical oceanography and climate dynamics.

Ocean circulation --- Differentiable dynamical systems. --- Mathematical models. --- El Niňo Current --- El Niño Current --- Oceanography. --- Electronic data processing. --- Climatology. --- Complex Systems. --- Numeric Computing. --- Atmospheric Sciences. --- Dynamical Systems and Ergodic Theory. --- ADP (Data processing) --- Automatic data processing --- Data processing --- EDP (Data processing) --- IDP (Data processing) --- Integrated data processing --- Computers --- Office practice --- Oceanography, Physical --- Oceanology --- Physical oceanography --- Thalassography --- Earth sciences --- Marine sciences --- Ocean --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Automation --- Mathematical geography. --- Statistical physics. --- Dynamical systems. --- Numerical analysis. --- Atmospheric sciences. --- Dynamics. --- Ergodic theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Atmospheric sciences --- Atmosphere --- Mathematical analysis --- Mathematical statistics --- Climate --- Climate science --- Climate sciences --- Science of climate --- Atmospheric science --- Statistical methods --- Corriente del Niño --- Holy Child Current --- Niño Current --- Ocean circulation - Mathematical models. --- El Niňo Current - Mathematical models.

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This work presents a unified treatment of three important integrable problems relevant to both Celestial and Quantum Mechanics. Under discussion are the Kepler (two-body) problem and the Euler (two-fixed center) problem, the latter being the more complex and more instructive, as it exhibits a richer and more varied solution structure. Further, because of the interesting investigations by the 20th century mathematical physicist J.P. Vinti, the Euler problem is now recognized as being intimately linked to the Vinti (Earth-satellite) problem. Here the analysis of these problems is shown to follow a definite shared pattern yielding exact forms for the solutions. A central feature is the detailed treatment of the planar Euler problem where the solutions are expressed in terms of Jacobian elliptic functions, yielding analytic representations for the orbits over the entire parameter range. This exhibits the rich and varied solution patterns that emerge in the Euler problem, which are illustrated in the appendix. A comparably detailed analysis is performed for the Earth-satellite (Vinti) problem. Key features: * Highlights shared features in the unified treatment of the Kepler, Euler, and Vinti problems * Raises challenges in analysis and astronomy, placing this trio of problems in the modern context * Includes a full analysis of the planar Euler problem * Highlights the complex and surprising orbit patterns that arise from the Euler problem * Provides a detailed analysis and solution for the Earth-satellite problem The analysis and results in this work will be of interest to graduate students in mathematics and physics (including physical chemistry) and researchers concerned with the general areas of dynamical systems, statistical mechanics, and mathematical physics and has direct application to celestial mechanics, astronomy, orbital mechanics, and aerospace engineering.

Mathematics. --- Applications of Mathematics. --- Astronomy, Astrophysics and Cosmology. --- Mathematical Methods in Physics. --- Statistical Physics. --- Mechanics. --- Dynamical Systems and Ergodic Theory. --- Differentiable dynamical systems. --- Mathematical physics. --- Statistical physics. --- Astronomy. --- Mathématiques --- Dynamique différentiable --- Physique mathématique --- Physique statistique --- Mécanique --- Astronomie --- Celestial mechanics. --- Two-body problem. --- Celestial mechanics --- Two-body problem --- Astronomy & Astrophysics --- Physical Sciences & Mathematics --- Theoretical Astronomy --- Problem of two bodies --- Gravitational astronomy --- Mechanics, Celestial --- Physics. --- Dynamics. --- Ergodic theory. --- Astrophysics. --- Cosmology. --- Dynamical systems. --- Theoretical, Mathematical and Computational Physics. --- Statistical Physics, Dynamical Systems and Complexity. --- Mechanics, Analytic --- Astrophysics --- Mechanics --- Classical Mechanics. --- Complex Systems. --- Physical mathematics --- Physics --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Mathematics --- Astronomical physics --- Astronomy --- Cosmic physics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamical systems --- Kinetics --- Force and energy --- Statics --- Mathematical statistics --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Statistical methods

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Many problems in theoretical economics are mathematically formalized as dynam­ ical systems of difference and differential equations. In recent years a truly open approach to studying the dynamical behavior of these models has begun to make its way into the mainstream. That is, economists formulate their hypotheses and study the dynamics of the resulting models rather than formulating the dynamics and studying hypotheses that could lead to models with such dynamics. This is a great progress over using linear models, or using nonlinear models with a linear approach, or even squeezing economic models into well-studied nonlinear systems from other fields. There are today a number of economic journals open to publishing this type of work and some of these have become important. There are several societies which have annual meetings on the subject and participation at these has been growing at a good rate. And of course there are methods and techniques avail­ able to a more general audience, as well as a greater availability of software for numerical and graphical analysis that makes this type of research even more excit­ ing. The lecturers for the Advanced School on Nonlinear Dynamical Systems in Economics, who represent a wide selection of the research areas to which the the­ ory has been applied, agree on the importance of simulations and computer-based analysis. The School emphasized computer applications of models and methods, and all contributors ran computer lab sessions.

Economics/Management Science. --- Economics/Management Science, general. --- Economic Growth. --- Economic Theory. --- Macroeconomics/Monetary Economics. --- Economics. --- Endogenous growth (Economics). --- Macroeconomics. --- Economie politique --- Croissance endogène (Economie politique) --- Macroéconomie --- Chaotic behavior in systems. --- Differentiable dynamical systems. --- Economics, Mathematical. --- Nonlinear theories. --- Business & Economics --- Economic Theory --- Nonlinear problems --- Nonlinearity (Mathematics) --- Economics --- Mathematical economics --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Mathematics --- Economic theory. --- Economic growth. --- Economic Theory/Quantitative Economics/Mathematical Methods. --- Macroeconomics/Monetary Economics//Financial Economics. --- Calculus --- Mathematical analysis --- Mathematical physics --- Econometrics --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- System theory --- Methodology --- Development, Economic --- Economic growth --- Growth, Economic --- Economic policy --- Statics and dynamics (Social sciences) --- Development economics --- Resource curse --- Economic theory --- Political economy --- Social sciences --- Economic man

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The Markov chain approximation methods are widely used for the numerical solution of nonlinear stochastic control problems in continuous time. This book extends the methods to stochastic systems with delays. Because such problems are infinite-dimensional, many new issues arise in getting good numerical approximations and in the convergence proofs. Useful forms of numerical algorithms and system approximations are developed in this work, and the convergence proofs are given. All of the usual cost functions are treated as well as singular and impulsive controls. A major concern is on representations and approximations that use minimal memory. Features and topics include: * Surveys properties of the most important stochastic dynamical models, including singular control, and those for diffusion and reflected diffusion models. * Gives approximations to the dynamical models that simplify the numerical problem, but have only small effects on the behavior. * Develops an ergodic theory for reflected diffusions with delays, as well as model simplifications useful for numerical approximations for average cost per unit time problems. * Provides numerical algorithms for models with delays in the path, or path and control, with reduced memory requirements. * Develops transformations of the problem that yield more efficient approximations when the control, driving Wiener process, and/or reflection processes might be delayed, as well as the path. * Presents examples with applications to control and modern communications systems. The book is the first on the subject and will be of interest to all those who work with stochastic delay equations and whose main interest is in either the use of the algorithms or the underlying mathematics. An excellent resource for graduate students, researchers, and practitioners, the work may be used as a graduate-level textbook for a special topics course or seminar on numerical methods in stochastic control.

Mathematics. --- Systems Theory, Control. --- Numerical Analysis. --- Operations Research, Mathematical Programming. --- Probability Theory and Stochastic Processes. --- Dynamical Systems and Ergodic Theory. --- Computational Intelligence. --- Differentiable dynamical systems. --- Systems theory. --- Numerical analysis. --- Operations research. --- Distribution (Probability theory). --- Engineering. --- Mathématiques --- Dynamique différentiable --- Analyse numérique --- Recherche opérationnelle --- Distribution (Théorie des probabilités) --- Ingénierie --- Stochastic systems. --- Stochastic systems --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Systems, Stochastic --- Mathematical analysis. --- Analysis (Mathematics). --- Dynamics. --- Ergodic theory. --- System theory. --- Management science. --- Probabilities. --- Analysis. --- Operations Research, Management Science. --- Stochastic processes --- System analysis --- Distribution (Probability theory. --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Philosophy --- 517.1 Mathematical analysis --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Quantitative business analysis --- Management --- Problem solving --- Operations research --- Statistical decision --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory