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Group theory --- Differential equations --- Attractors (Mathematics) --- Lyapunov exponents. --- Stokes equations. --- Attracteurs (Mathématiques) --- Liapounov, Exposants de --- Equations de Stokes --- 51 <082.1> --- Mathematics--Series --- Attracteurs (Mathématiques) --- Navier-Stokes, Équations de. --- Liapounov, Exposants de. --- Attracteurs (mathématiques) --- Lyapunov exponents --- Stokes equations --- Stokes differential equations --- Stokes's differential equations --- Stokes's equations --- Differential equations, Partial --- Liapunov exponents --- Lyapunov characteristic exponents --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Differentiable dynamical systems

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This monograph offers a coherent, self-contained account of the theory of Sinai–Ruelle–Bowen measures and decay of correlations for nonuniformly hyperbolic dynamical systems. A central topic in the statistical theory of dynamical systems, the book in particular provides a detailed exposition of the theory developed by L.-S. Young for systems admitting induced maps with certain analytic and geometric properties. After a brief introduction and preliminary results, Chapters 3, 4, 6 and 7 provide essentially the same pattern of results in increasingly interesting and complicated settings. Each chapter builds on the previous one, apart from Chapter 5 which presents a general abstract framework to bridge the more classical expanding and hyperbolic systems explored in Chapters 3 and 4 with the nonuniformly expanding and partially hyperbolic systems described in Chapters 6 and 7. Throughout the book, the theory is illustrated with applications. A clear and detailed account of topics of current research interest, this monograph will be of interest to researchers in dynamical systems and ergodic theory. In particular, beginning researchers and graduate students will appreciate the accessible, self-contained presentation.

Dynamics. --- Ergodic theory. --- Vibration. --- Dynamical systems. --- Dynamical Systems and Ergodic Theory. --- Vibration, Dynamical Systems, Control. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Cycles --- Sound --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Attractors (Mathematics) --- Differential equations, Hyperbolic. --- Hyperbolic differential equations --- Differential equations, Partial --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Differentiable dynamical systems

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"Hyperbolic Chaos: A Physicist’s View” presents recent progress on uniformly hyperbolic attractors in dynamical systems from a physical rather than mathematical perspective (e.g. the Plykin attractor, the Smale – Williams solenoid). The structurally stable attractors manifest strong stochastic properties, but are insensitive to variation of functions and parameters in the dynamical systems. Based on these characteristics of hyperbolic chaos, this monograph shows how to find hyperbolic chaotic attractors in physical systems and how to design a physical systems that possess hyperbolic chaos.   This book is designed as a reference work for university professors and researchers in the fields of physics, mechanics, and engineering.   Dr. Sergey P. Kuznetsov is a professor at the Department of Nonlinear Processes, Saratov State University, Russia.  .

Cosmology. --- Geometry, Hyperbolic. --- Quantum chaos. --- Mathematical physics --- Physics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Atomic Physics --- Applied Physics --- Differential equations, Hyperbolic. --- Attractors (Mathematics) --- Physics. --- Natural philosophy --- Philosophy, Natural --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Hyperbolic differential equations --- System theory. --- Statistical physics. --- Vibration. --- Dynamical systems. --- Dynamics. --- Nonlinear Dynamics. --- Systems Theory, Control. --- Vibration, Dynamical Systems, Control. --- Physical sciences --- Dynamics --- Differentiable dynamical systems --- Differential equations, Partial --- Systems theory. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Cycles --- Mechanics --- Sound --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Statics --- Systems, Theory of --- Systems science --- Science --- Mathematical statistics --- Philosophy --- Statistical methods

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This work focuses on the preservation of attractors and saddle points of ordinary differential equations under discretisation. In the 1980s, key results for autonomous ordinary differential equations were obtained – by Beyn for saddle points and by Kloeden & Lorenz for attractors. One-step numerical schemes with a constant step size were considered, so the resulting discrete time dynamical system was also autonomous. One of the aims of this book is to present new findings on the discretisation of dissipative nonautonomous dynamical systems that have been obtained in recent years, and in particular to examine the properties of nonautonomous omega limit sets and their approximations by numerical schemes – results that are also of importance for autonomous systems approximated by a numerical scheme with variable time steps, thus by a discrete time nonautonomous dynamical system.

Attractors (Mathematics) --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Mathematics. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Numerical analysis. --- Numerical Analysis. --- Dynamical Systems and Ergodic Theory. --- Ordinary Differential Equations. --- Differentiable dynamical systems --- Differentiable dynamical systems. --- Differential Equations. --- 517.91 Differential equations --- Differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Mathematical analysis --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics

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This book treats the theory of pullback attractors for non-autonomous dynamical systems. While the emphasis is on infinite-dimensional systems, the results are also applied to a variety of finite-dimensional examples.   The purpose of the book is to provide a summary of the current theory, starting with basic definitions and proceeding all the way to state-of-the-art results. As such it is intended as a primer for graduate students, and a reference for more established researchers in the field.   The basic topics are existence results for pullback attractors, their continuity under perturbation, techniques for showing that their fibres are finite-dimensional, and structural results for pullback attractors for small non-autonomous perturbations of gradient systems (those with a Lyapunov function).  The structural results stem from a dynamical characterisation of autonomous gradient systems, which shows in particular that such systems are stable under perturbation. Application of the structural results relies on the continuity of unstable manifolds under perturbation, which in turn is based on the robustness of exponential dichotomies: a self-contained development of  these topics is given in full. After providing all the necessary theory the book treats a number of model problems in detail, demonstrating the wide applicability of the definitions and techniques introduced: these include a simple Lotka-Volterra ordinary differential equation, delay differential equations, the two-dimensional Navier-Stokes equations, general reaction-diffusion problems, a non-autonomous version of the Chafee-Infante problem, a comparison of attractors in problems with perturbations to the diffusion term, and a non-autonomous damped wave equation. Alexandre N. Carvalho is a Professor at the University of Sao Paulo, Brazil. José A. Langa is a Profesor Titular at the University of Seville, Spain. James C. Robinson is a Professor at the University of Warwick, UK.

Attractors (Mathematics). --- Differentiable dynamical systems. --- Differential equations, Partial. --- Mathematics. --- Attractors (Mathematics) --- Natural Science Disciplines --- Disciplines and Occupations --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Calculus --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics) --- Dynamics. --- Ergodic theory. --- Partial differential equations. --- Manifolds (Mathematics). --- Complex manifolds. --- Partial Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Partial differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Math --- Science --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Differentiable dynamical systems --- Differential equations, partial. --- Cell aggregation --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Cell aggregation -- Mathematics. --- Manifolds and Cell Complexes (incl. Diff. Topology)