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This invaluable book examines qualitative and quantitative methods for nonlinear differential equations, as well as integrability and nonintegrability theory. Starting from the idea of a constant of motion for simple systems of differential equations, it investigates the essence of integrability, its geometrical relevance and dynamical consequences. Integrability theory is approached from different perspectives, first in terms of differential algebra, then in terms of complex time singularities and finally from the viewpoint of phase geometry (for both Hamiltonian and non-Hamiltonian systems)

Differentiable dynamical systems. --- Differential equations, Nonlinear. --- Nonlinear differential equations --- Nonlinear theories --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics

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A systematic and comprehensive introduction to the study of nonlinear dynamical systems, in both discrete and continuous time, for nonmathematical students and researchers working in applied fields. An understanding of linear systems and the classical theory of stability are essential although basic reviews of the relevant material are provided. Further chapters are devoted to the stability of invariant sets, bifurcation theory, chaotic dynamics and the transition to chaos. In the final two chapters the authors approach the subject from a measure-theoretical point of view and compare results to those given for the geometrical or topological approach of the first eight chapters. Includes about one hundred exercises. A Windows-compatible software programme called DMC, provided free of charge through a website dedicated to the book, allows readers to perform numerical and graphical analysis of dynamical systems. Also available on the website are computer exercises and solutions to selected book exercises. See www.cambridge.org/economics/resources

Operational research. Game theory --- Differentiable dynamical systems. --- Nonlinear theories. --- Differentiable dynamical systems --- Nonlinear theories --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- 515.352 --- Nonlinear problems --- Nonlinearity (Mathematics) --- Calculus --- Mathematical analysis --- Mathematical physics --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Dynamique différentiable --- Théories non linéaires --- Systèmes dynamiques --- Business, Economy and Management --- Economics

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Differential equations --- Classical mechanics. Field theory --- Differentiable dynamical systems --- Lagrange equations --- Variational principles --- Extremum principles --- Minimal principles --- Variation principles --- Calculus of variations --- D'Alembert equation --- Equations, Euler-Lagrange --- Equations, Lagrange --- Euler-Lagrange equations --- Lagrangian equations --- Equations of motion --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Lagrange equations. --- Lagrange, Équations de --- Differentiable dynamical systems. --- Systèmes dynamiques --- Variational principles. --- Principes variationnels --- Lagrange, Équations de. --- Systèmes dynamiques. --- Principes variationnels.

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This book summarizes and highlights progress in our understanding of Dy­ namical Systems during six years of the German Priority Research Program "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems" . The program was funded by the Deutsche Forschungsgemeinschaft (DFG) and aimed at combining, focussing, and enhancing research efforts of active groups in the field by cooperation on a federal level. The surveys in the book are addressed to experts and non-experts in the mathematical community alike. In addition they intend to convey the significance of the results for applications far into the neighboring disciplines of Science. Three fundamental topics in Dynamical Systems are at the core of our research effort: behavior for large time dimension measure, and chaos Each of these topics is, of course, a highly complex problem area in itself and does not fit naturally into the deplorably traditional confines of any of the disciplines of ergodic theory, analysis, or numerical analysis alone. The necessity of mathematical cooperation between these three disciplines is quite obvious when facing the formidahle task of establishing a bidirectional transfer which bridges the gap between deep, detailed theoretical insight and relevant, specific applications. Both analysis and numerical analysis playa key role when it comes to huilding that bridge. Some steps of our joint bridging efforts are collected in this volume. Neither our approach nor the presentations in this volume are monolithic.

Differentiable dynamical systems --- Ordinary differential equations --- 519.2 --- Ergodic theory --- Mathematical analysis --- 681.3*I6 --- 681.3*I6 Simulation and modeling (Computing methodologies)--See also {681.3*G3} --- Simulation and modeling (Computing methodologies)--See also {681.3*G3} --- 517.1 Mathematical analysis --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- 519.2 Probability. Mathematical statistics --- Probability. Mathematical statistics --- Differentiable dynamical systems. --- Ergodic theory. --- Mathematical analysis. --- Dynamique différentiable --- Théorie ergodique --- Analyse mathématique --- 517.1. --- Applied mathematics. --- Engineering mathematics. --- Analysis (Mathematics). --- Statistical physics. --- Dynamical systems. --- Probabilities. --- Applications of Mathematics. --- Analysis. --- Complex Systems. --- Probability Theory and Stochastic Processes. --- Statistical Physics and Dynamical Systems. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Engineering --- Engineering analysis --- Statistical methods --- 517.1