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This book presents a detailed derivation of the spectral properties of the Recursion Operators allowing one to derive all the fundamental properties of the soliton equations and to study their Hamiltonian hierarchies. Thus it is demonstrated that the inverse scattering method for solving soliton equations is a nonlinear generalization of the Fourier transform. The book brings together the spectral and the geometric approaches and as such will be useful to a wide readership: from researchers in the field of nonlinear completely integrable evolution equations to graduate and post-graduate students.

Hamiltonian systems. --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- Mathematical physics. --- Global analysis (Mathematics). --- Geometry. --- Physics. --- Mathematical Methods in Physics. --- Analysis. --- Theoretical, Mathematical and Computational Physics. --- Physics, general. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Mathematics --- Euclid's Elements --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Physical mathematics --- Physics --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis --- Mathematical analysis

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In the analysis and synthesis of contemporary systems, engineers and scientists are frequently confronted with increasingly complex models that may simultaneously include components whose states evolve along continuous time and discrete instants; components whose descriptions may exhibit nonlinearities, time lags, transportation delays, hysteresis effects, and uncertainties in parameters; and components that cannot be described by various classical equations, as in the case of discrete-event systems, logic commands, and Petri nets. The qualitative analysis of such systems requires results for finite-dimensional and infinite-dimensional systems; continuous-time and discrete-time systems; continuous continuous-time and discontinuous continuous-time systems; and hybrid systems involving a mixture of continuous and discrete dynamics. Filling a gap in the literature, this textbook presents the first comprehensive stability analysis of all the major types of system models described above. Throughout the book, the applicability of the developed theory is demonstrated by means of many specific examples and applications to important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, artificial neural networks (with and without time delays), digital signal processing, a class of discrete-event systems (with applications to manufacturing and computer load balancing problems) and a multicore nuclear reactor model. The book covers the following four general topics: * Representation and modeling of dynamical systems of the types described above* Presentation of Lyapunov and Lagrange stability theory for dynamical systems defined on general metric spaces* Specialization of this stability theory to finite-dimensional dynamical systems* Specialization of this stability theory to infinite-dimensional dynamical systems Replete with exercises and requiring basic knowledge of linear algebra, analysis, and differential equations, the work may be used as a textbook for graduate courses in stability theory of dynamical systems. The book may also serve as a self-study reference for graduate students, researchers, and practitioners in applied mathematics, engineering, computer science, physics, chemistry, biology, and economics. [Publisher]

Differentiable dynamical systems --- Stability --- Differentiable dynamical systems. --- Stability. --- Dynamique différentiable --- Systèmes dynamiques --- Stabilité --- 517.9 --- Dynamics --- Mechanics --- Motion --- Vibration --- Benjamin-Feir instability --- Equilibrium --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Dynamique différentiable. --- Systèmes dynamiques. --- Stabilité.

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The main objective of this book is to give a broad unified introduction to the study of dimension and recurrence in hyperbolic dynamics. It includes the discussion of the foundations, main results, and main techniques in the rich interplay of four main areas of research: hyperbolic dynamics, dimension theory, multifractal analysis, and quantitative recurrence. It also gives a panorama of several selected topics of current research interest. More than half of the material appears here for the first time in book form, describing many recent developments in the area such as topics on irregular sets, variational principles, applications to number theory, measures of maximal dimension, multifractal nonrigidity, and quantitative recurrence. All the results are included with detailed proofs, many of them simplified or rewritten on purpose for the book. The text is self-contained and directed to researchers as well as graduate students that wish to have a global view of the theory together with a working knowledge of its main techniques. It will also be useful as as basis for graduate courses in dimension theory of dynamical systems, multifractal analysis, and pointwise dimension and recurrence in hyperbolic dynamics.

Differentiable dynamical systems. --- Hyperbolic groups. --- Dimension theory (Topology) --- Topology --- Group theory --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Cell aggregation --- Global analysis (Mathematics). --- Dynamical Systems and Ergodic Theory. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Analysis. --- Mathematics. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Dynamics. --- Ergodic theory. --- Manifolds (Mathematics). --- Complex manifolds. --- Mathematical analysis. --- Analysis (Mathematics). --- 517.1 Mathematical analysis --- Mathematical analysis --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- 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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Sequential Dynamical Systems (SDS) are a class of discrete dynamical systems which significantly generalize many aspects of systems such as cellular automata, and provide a framework for studying dynamical processes over graphs. This text is the first to provide a comprehensive introduction to SDS. Driven by numerous examples and thought-provoking problems, the presentation offers good foundational material on finite discrete dynamical systems which leads systematically to an introduction of SDS. Techniques from combinatorics, algebra and graph theory are used to study a broad range of topics, including reversibility, the structure of fixed points and periodic orbits, equivalence, morphisms and reduction. Unlike other books that concentrate on determining the structure of various networks, this book investigates the dynamics over these networks by focusing on how the underlying graph structure influences the properties of the associated dynamical system. This book is aimed at graduate students and researchers in discrete mathematics, dynamical systems theory, theoretical computer science, and systems engineering who are interested in analysis and modeling of network dynamics as well as their computer simulations. Prerequisites include knowledge of calculus and basic discrete mathematics. Some computer experience and familiarity with elementary differential equations and dynamical systems are helpful but not necessary.

Differentiable dynamical systems. --- Sequential analysis. --- Mathematical statistics --- Statistical decision --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Global analysis (Mathematics). --- Computer simulation. --- Mathematics. --- Computational complexity. --- Analysis. --- Dynamical Systems and Ergodic Theory. --- Simulation and Modeling. --- Applications of Mathematics. --- Discrete Mathematics in Computer Science. --- Complexity, Computational --- Electronic data processing --- Machine theory --- Math --- Science --- Computer modeling --- Computer models --- Modeling, Computer --- Models, Computer --- Simulation, Computer --- Electromechanical analogies --- Mathematical models --- Simulation methods --- Model-integrated computing --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- Dynamics. --- Ergodic theory. --- Applied mathematics. --- Engineering mathematics. --- Computer science—Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- 517.1 Mathematical analysis

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Nonlinear differential or difference equations are encountered not only in mathematics, but also in many areas of physics (evolution equations, propagation of a signal in an optical fiber), chemistry (reaction-diffusion systems), and biology (competition of species). This book introduces the reader to methods allowing one to build explicit solutions to these equations. A prerequisite task is to investigate whether the chances of success are high or low, and this can be achieved without any a priori knowledge of the solutions, with a powerful algorithm presented in detail called the Painlevé test. If the equation under study passes the Painlevé test, the equation is presumed integrable. If on the contrary the test fails, the system is nonintegrable or even chaotic, but it may still be possible to find solutions. The examples chosen to illustrate these methods are mostly taken from physics. These include on the integrable side the nonlinear Schrödinger equation (continuous and discrete), the Korteweg-de Vries equation, the Hénon-Heiles Hamiltonians, on the nonintegrable side the complex Ginzburg-Landau equation (encountered in optical fibers, turbulence, etc), the Kuramoto-Sivashinsky equation (phase turbulence), the Kolmogorov-Petrovski-Piskunov equation (KPP, a reaction-diffusion model), the Lorenz model of atmospheric circulation and the Bianchi IX cosmological model. Written at a graduate level, the book contains tutorial text as well as detailed examples and the state of the art on some current research.

Painlevé equations. --- Mathematical physics. --- Physical mathematics --- Physics --- Equations, Painlevé --- Functions, Painlevé --- Painlevé functions --- Painlevé transcendents --- Transcendents, Painlevé --- Differential equations, Nonlinear --- Mathematics --- Differentiable dynamical systems. --- Differential Equations. --- Differential equations, partial. --- Engineering mathematics. --- Chemistry --- Mathematical Methods in Physics. --- Dynamical Systems and Ergodic Theory. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Mathematical and Computational Engineering. --- Math. Applications in Chemistry. --- Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Physics. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Partial differential equations. --- Applied mathematics. --- Chemometrics. --- Chemistry, Analytic --- Analytical chemistry --- 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 --- Measurement --- Statistical methods