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This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey "Principal Structures? of Volume 1A.The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations).. Written by experts in the field.. The coverage of ergodic theory in these two parts o
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The Hungarian born mathematical genius, John von Neumann, was undoubtedly one of the greatest and most influential scientific minds of the 20th century. Von Neumann made fundamental contributions to Computing and he had a keen interest in Dynamical Systems, specifically Hydrodynamic Turbulence. This book, offering a state-of-the-art collection of papers in computational dynamical systems, is dedicated to the memory of von Neumann. Including contributions from J E Marsden, P J Holmes, M Shub, A Iserles, M Dellnitz and J Guckenheimer, this book offers a unique combination of theoretical and app
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Stochastic processes --- Differentiable dynamical systems --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Random processes --- Probabilities
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This book is a unique blend of difference equations theory and its exciting applications to economics. It deals with not only theory of linear (and linearized) difference equations, but also nonlinear dynamical systems which have been widely applied to economic analysis in recent years. It studies most important concepts and theorems in difference equations theory in a way that can be understood by anyone who has basic knowledge of calculus and linear algebra. It contains well-known applications and many recent developments in different fields of economics. The book also simulates many models
Economics, Mathematical. --- Difference equations. --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Calculus of differences --- Differences, Calculus of --- Equations, Difference --- Economics --- Mathematical economics --- Econometrics --- Mathematics --- Methodology
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This book presents a new theory on the transition to dynamical chaos for two-dimensional nonautonomous, and three-dimensional, many-dimensional and infinitely-dimensional autonomous nonlinear dissipative systems of differential equations including nonlinear partial differential equations and differential equations with delay arguments. The transition is described from the Feigenbaum cascade of period doubling bifurcations of the original singular cycle to the complete or incomplete Sharkovskii subharmonic cascade of bifurcations of stable limit cycles with arbitrary period and finally to the
Differentiable dynamical systems. --- Differential equations. --- Dynamics. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- 517.91 Differential equations --- Differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics
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The book deals essentially with systems of polynomial autonomous ordinary differential equations in two real variables. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects as a thorough study of the center/focus problem and recent results on integrability. In the last two chapters the performant software tool P4 is introduced: based on both algebraic manipulation and numerical calculation, this was conceived for the purpose of drawing "Polynomial Planar Phase Portraits" on part of the plane, or on a Poincaré compactification, or even on a Poincaré-Lyapunov compactification of the plane. From the start, differential systems are represented by vector fields enabling, in full strength, a dynamical systems approach. All essential notions, including invariant manifolds, normal forms, desingularization of singularities, index theory and limit cycles, are introduced and the main results are proved for smooth systems with the necessary specifications for analytic and polynomial systems. The book is very appropriate for a first course in dynamical systems, presenting the basic notions in the study of individual two dimensional systems. Not only does it provide simple and appropriate proofs, but it also contains a lot of exercises and presents a survey of interesting results with the necessary references to the literature.
Differential equations --- Qualitative theory. --- Numerical solutions. --- 517.91 Differential equations --- Differential Equations. --- Differentiable dynamical systems. --- Ordinary Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Differential equations. --- 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
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This book is the first monograph devoted exclusively to strange nonchaotic attractors (SNA), recently discovered objects with a special kind of dynamical behavior between order and chaos in dissipative nonlinear systems under quasiperiodic driving. A historical review of the discovery and study of SNA, mathematical and physically-motivated examples, and a review of known experimental studies of SNA are presented. The main focus is on the theoretical analysis of strange nonchaotic behavior by means of different tools of nonlinear dynamics and statistical physics (bifurcation analysis, Lyapunov
Attractors (Mathematics) --- Chaotic behavior in systems. --- Differentiable dynamical systems. --- Nonlinear systems. --- Systems, Nonlinear --- System theory --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- Attracting sets (Mathematics) --- Attractors of a dynamical system --- Dynamical system, Attractors of --- Sets, Attracting (Mathematics)
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This important book reviews applications of optimization and optimal control theory to modern problems in physics, nano-science and finance. The theory presented here can be efficiently applied to various problems, such as the determination of the optimal shape of a laser pulse to induce certain excitations in quantum systems, the optimal design of nanostructured materials and devices, or the control of chaotic systems and minimization of the forecast error for a given forecasting model (for example, artificial neural networks). Starting from a brief review of the history of variational calcul
Chaotic behavior in systems. --- Control theory. --- Differentiable dynamical systems. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Dynamics --- Machine theory --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Nonlinear theories --- System theory
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The notion of a ?xed point plays a crucial role in numerous branches of mat- maticsand its applications. Informationabout the existence of such pointsis often the crucial argument in solving a problem. In particular, topological methods of ?xed point theory have been an increasing focus of interest over the last century. These topological methods of ?xed point theory are divided, roughly speaking, into two types. The ?rst type includes such as the Banach Contraction Principle where the assumptions on the space can be very mild but a small change of the map can remove the ?xed point. The second type, on the other hand, such as the Brouwer and Lefschetz Fixed Point Theorems, give the existence of a ?xed point not only for a given map but also for any its deformations. This book is an exposition of a part of the topological ?xed and periodic point theory, of this second type, based on the notions of Lefschetz and Nielsen numbers. Since both notions are homotopyinvariants, the deformationis used as an essential method, and the assertions of theorems typically state the existence of ?xed or periodic points for every map of the whole homotopy class, we refer to them as homotopy methods of the topological ?xed and periodic point theory.
Fixed point theory. --- Homotopy theory. --- Deformations, Continuous --- Topology --- Fixed point theorems (Topology) --- Nonlinear operators --- Coincidence theory (Mathematics) --- Algebraic topology. --- Differentiable dynamical systems. --- Algebraic Topology. --- Dynamical Systems and Ergodic Theory. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- 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
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The study of dynamical systems is a well established field. The authors have written this book in an attempt to provide a panorama of several aspects, that are of interest to mathematicians and physicists alike. The book collects the material of several courses at the graduate level given by the authors. Thus, the exposition avoids detailed proofs in exchange for numerous illustrations and examples, while still maintaining sufficient precision. Apart from common subjects in this field, a lot of attention is given to questions of physical measurement and stochastic properties of chaotic 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 physics. --- Dynamical Systems and Ergodic Theory. --- Mathematical Methods in Physics. --- Physical mathematics --- Physics --- Mathematics --- Dynamics. --- Ergodic theory. --- Physics. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics
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