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1 online resource (384 pages) |
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text txt rdacontent |
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computer c rdamedia |
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online resource cr rdacarrier |
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Series On Number Theory And Its Applications ; v.58 |
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Series On Number Theory And Its Applications
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| Note |
Intro -- Contents -- Preface -- 1. Systems of Ordinary Differential Equations -- 1.1 Basic Definitions and Theorems -- 1.1.1 Fields of directions and their integral curves -- 1.1.2 Vector fields, differential equations, integral and phase curves -- 1.1.3 Theorems of existence and uniqueness of solutions -- 1.1.4 Differentiable dependence of solutions from initial conditions and parameters, the equations in variations -- 1.1.5 Dissipative and conservative systems of differential equations -- 1.1.6 Numerical methods for solution of systems of ordinary differential equations -- 1.1.7 Ill-posedness of numerical methods in solution of systems of ordinary differential equations -- 1.2 Singular Points and Their Invariant Manifolds -- 1.2.1 Singular points of systems of ordinary differential equations -- 1.2.2 Stability of singular points and stationary solutions -- 1.2.3 Invariant manifolds -- 1.2.4 Singular points of linear vector fields -- 1.2.5 Separatrices of singular points, homoclinic and heteroclinic trajectories, separatrix contours -- 1.3 Periodic and Nonperiodic Solutions, Limit Cycles and Invariant Tori -- 1.3.1 Periodic solutions -- 1.3.2 Limit cycles -- 1.3.3 Poincare map -- 1.3.4 Invariant tori -- 1.4 Attractors of Dissipative Systems of Ordinary Differential Equations -- 1.4.1 Basic definitions -- 1.4.2 Classical regular attractors of dissipative systems of ordinary differential equations -- 1.4.3 Classical irregular attractors of dissipative dynamical systems -- 1.4.4 Dimension of attractors, fractals -- 2. Bifurcations in Nonlinear Systems of Ordinary Differential Equations -- 2.1 Structural Stability and Bifurcations -- 2.1.1 Structural stability -- 2.1.2 Bifurcations -- 2.2 One-Parametrical Local Bifurcations -- 2.2.1 Bifurcations of stable singular points -- 2.2.2 Bifurcations of stable limit cycles |
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2.2.3 Bifurcations of stable two-dimensional tori -- 2.3 The Simplest Two-Parametrical Local Bifurcations -- 2.3.1 The normal form of a fold -- 2.3.2 The normal form of an assembly -- 2.4 Nonlocal Bifurcations -- 2.4.1 Bifurcations of homoclinic separatrix contours -- 2.4.2 Bifurcations of heteroclinic separatrix contours -- 2.4.3 Approximate method for finding bifurcation points of separatrix contours of singular points -- 2.4.4 Cascades of bifurcations, scenarios of transition to chaos -- 2.4.5 Bifurcations of irregular (singular) attractors -- 3 Chaotic Systems of Ordinary Differential Equations -- 3.1 System of the Lorenz Equations -- 3.1.1 Classical scenario of birth of the Lorenz attractor -- 3.1.2 Scenario of birth of the Lorenz attractor through an incomplete double homoclinic cascade of bifurcations -- 3.1.3 Scenario of birth of a complete double homoclinic attractor in the Lorenz system -- 3.1.4 Bifurcations of homoclinic and heteroclinic contours in the Lorenz system -- 3.1.5 Diagrams of nonlocal bifurcations in the Lorenz system -- 3.2 The Complex System of Lorenz Equations -- 3.2.1 Scenario of transition to chaos -- 3.3 Systems of the Rossler Equations -- 3.4 The Chua System -- 3.5 Other Chaotic Systems of Ordinary Differential Equations -- 3.5.1 The Vallis systems -- 3.5.2 The Rikitaki system -- 3.5.3 The "Simple" system -- 3.5.4 The Rabinovich-Fabrikant system -- 3.6 Final Remarks and Conclusions -- 4. Principles of the Theory of Dynamical Chaos in Dissipative Systems of Ordinary Differential Equations -- 4.1 Theory of One-Dimensional Smooth Mappings -- 4.1.1 Monotonic invertible mappings -- 4.1.2 Nonmonotonic mappings -- 4.2 Feigenbaum Cascade of Period Doubling Bifurcations of Cycles of One-Dimensional Mappings -- 4.2.1 Logistic mapping -- 4.2.2 Period doubling operator -- 4.2.3 Feigenbaum universality |
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4.2.4 Dimension of the Feigenbaum attractor -- 4.3 Sharkovskii Subharmonic Cascade of Bifurcations of Cycles of One-Dimensional Mappings -- 4.3.1 The Sharkovskii's theorem -- 4.3.2 Behind the Feigenbaum cascade -- 4.4 Dynamical Chaos in Two-Dimensional Non-Autonomous Systems of Differential Equations -- 4.4.1 Rotor type singular points -- 4.4.2 Scenario of transition to chaos -- 4.4.3 Dynamical chaos in some classical two-dimensional non-autonomous systems -- 4.5 Dynamical Chaos in Three-Dimensional Autonomous Systems of Differential Equations -- 4.5.1 Singular cycles of three-dimensional autonomous systems -- 4.5.2 Singular attractors of three-dimensional autonomous systems -- 4.5.3 Some examples of three-dimensional autonomous systems with singular attractors -- 4.6 Final Remarks and Conclusions -- 5. Dynamical Chaos in Infinite-Dimensional Systems of Differential Equations -- 5.1 Regular Dynamics and Diffusion Chaos in Reaction-Diffusion Systems -- 5.1.1 Turing and Andronov-Hopf bifurcations in the Brusselator model -- 5.1.2 Diffusion chaos for the Brusselator in a ring -- 5.1.3 Diffusion chaos in the Brusselator on a segment -- 5.2 Transition to Spatio-Temporal Chaos in the Kuramoto-Tsuzuki Equation -- 5.2.1 Scenario of transition to chaos in system of few-mode approximations -- 5.2.2 Transition to chaos in the space of Fourier coefficients -- 5.2.3 Scenario of transition to chaos in the phase space of the Kuramoto-Tsuzuki equation -- 5.3 Dynamical Chaos in Differential Equations with Delay Argument -- 5.4 Cycles and Chaos in Distributed Economic Systems -- 5.4.1 Description of the model of self-developing market economy -- 5.4.2 Behavior of macroeconomic variables -- 5.4.3 Behavior of economic variables in the presence of diffusion of capital and consumer demand -- 6. Chaos Control in Systems of Differential Equations |
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6.1 Ott-Grebogi-Yorke and Pyragas methods -- 6.1.1 The OGY-method -- 6.1.2 The Pyragas method -- 6.2 The Magnitskii Method -- 6.2.1 Localization and stabilization of unstable fixed points and unstable cycles of chaotic mappings -- 6.2.2 Localization and stabilization of unstable fixed points of chaotic dynamical systems -- 6.2.3 Localization and stabilization of unstable cycles of chaotic dynamical systems -- 6.2.4 Chaos control in equations with delay argument -- 6.2.5 Stabilization of a thermodynamic branch in reaction-diffusion systems of equations -- 6.3 Reconstruction of Dynamical System on Trajectory of Irregular Attractor -- Bibliography -- Index |
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Key Features:Presents a new point of view on the nature of dynamical chaos in differential equationsDevelops a universal theory of dynamical chaos in nonlinear dissipative systems of differential equationsUncovers a rotor type singular point as a bridge between one-dimensional mappings and differential equationsDescribes a universal scenario of transition to dynamical and spatio-temporal chaosIncludes numerous analytical and numerical examples and illustrations |
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Description based on publisher supplied metadata and other sources |
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries |
| Link |
Print version: Magnit?s?kii?, N. A New Methods for Chaotic Dynamics
Singapore : World Scientific Publishing Co Pte Ltd,c2006 9789812568175
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| Subject |
Differentiable dynamical systems.;Differential equations.;Dynamics
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Electronic books
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| Alt Author |
Sidorov, Sergey Vasilevich
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