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Bifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems /
Based on a lecture course at the Ecole Polytechnique (Paris), this text gives a rigorous introduction to many of the key ideas in nonlinear analysis, dynamical systems and bifurcation theory including catastrophe theory. Wherever appropriate it emphasizes a geometrical or coordinate-free approach which allows a clear focus on the essential mathematical structures. Taking a unified view, it brings out features common to different branches of the subject while giving ample references for more advanced or technical developments.
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| Format: | Texto biblioteca |
| Language: | eng |
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Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer,
2000
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| Subjects: | Mathematics., Dynamics., Ergodic theory., Global analysis (Mathematics)., Manifolds (Mathematics)., Differential geometry., Differential Geometry., Global Analysis and Analysis on Manifolds., Dynamical Systems and Ergodic Theory., |
| Online Access: | http://dx.doi.org/10.1007/978-3-642-57134-3 |
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Mathematics. Dynamics. Ergodic theory. Global analysis (Mathematics). Manifolds (Mathematics). Differential geometry. Mathematics. Differential Geometry. Global Analysis and Analysis on Manifolds. Dynamical Systems and Ergodic Theory. Mathematics. Dynamics. Ergodic theory. Global analysis (Mathematics). Manifolds (Mathematics). Differential geometry. Mathematics. Differential Geometry. Global Analysis and Analysis on Manifolds. Dynamical Systems and Ergodic Theory. |
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Mathematics. Dynamics. Ergodic theory. Global analysis (Mathematics). Manifolds (Mathematics). Differential geometry. Mathematics. Differential Geometry. Global Analysis and Analysis on Manifolds. Dynamical Systems and Ergodic Theory. Mathematics. Dynamics. Ergodic theory. Global analysis (Mathematics). Manifolds (Mathematics). Differential geometry. Mathematics. Differential Geometry. Global Analysis and Analysis on Manifolds. Dynamical Systems and Ergodic Theory. Demazure, Michel. author. SpringerLink (Online service) Bifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems / |
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Based on a lecture course at the Ecole Polytechnique (Paris), this text gives a rigorous introduction to many of the key ideas in nonlinear analysis, dynamical systems and bifurcation theory including catastrophe theory. Wherever appropriate it emphasizes a geometrical or coordinate-free approach which allows a clear focus on the essential mathematical structures. Taking a unified view, it brings out features common to different branches of the subject while giving ample references for more advanced or technical developments. |
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Texto |
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Mathematics. Dynamics. Ergodic theory. Global analysis (Mathematics). Manifolds (Mathematics). Differential geometry. Mathematics. Differential Geometry. Global Analysis and Analysis on Manifolds. Dynamical Systems and Ergodic Theory. |
| author |
Demazure, Michel. author. SpringerLink (Online service) |
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Demazure, Michel. author. SpringerLink (Online service) |
| author_sort |
Demazure, Michel. author. |
| title |
Bifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems / |
| title_short |
Bifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems / |
| title_full |
Bifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems / |
| title_fullStr |
Bifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems / |
| title_full_unstemmed |
Bifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems / |
| title_sort |
bifurcations and catastrophes [electronic resource] : geometry of solutions to nonlinear problems / |
| publisher |
Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, |
| publishDate |
2000 |
| url |
http://dx.doi.org/10.1007/978-3-642-57134-3 |
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AT demazuremichelauthor bifurcationsandcatastropheselectronicresourcegeometryofsolutionstononlinearproblems AT springerlinkonlineservice bifurcationsandcatastropheselectronicresourcegeometryofsolutionstononlinearproblems |
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1756268153708478464 |
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KOHA-OAI-TEST:2057442018-07-30T23:35:11ZBifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems / Demazure, Michel. author. SpringerLink (Online service) textBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer,2000.engBased on a lecture course at the Ecole Polytechnique (Paris), this text gives a rigorous introduction to many of the key ideas in nonlinear analysis, dynamical systems and bifurcation theory including catastrophe theory. Wherever appropriate it emphasizes a geometrical or coordinate-free approach which allows a clear focus on the essential mathematical structures. Taking a unified view, it brings out features common to different branches of the subject while giving ample references for more advanced or technical developments.1. Local Inversion -- 1.1 Introduction -- 1.2 A Preliminary Statement -- 1.3 Partial Derivatives. Strictly Differentiable Functions -- 1.4 The Local Inversion Theorem: General Statement -- 1.5 Functions of Class Cr -- 1.6 The Local Inversion Theorem for Crmaps -- 1.7 Curvilinear Coordinates -- 1.8 Generalizations of the Local Inversion Theorem -- 2. Submanifolds -- 2.1 Introduction -- 2.2 Definitions of Submanifolds -- 2.3 First Examples -- 2.4 Tangent Spaces of a Submanifold -- 2.5 Transversality: Intersections -- 2.6 Transversality: Inverse Images -- 2.7 The Implicit Function Theorem -- 2.8 Diffeomorphisms of Submanifolds -- 2.9 Parametrizations, Immersions and Embeddings -- 2.10 Proper Maps; Proper Embeddings -- 2.11 From Submanifolds to Manifolds -- 2.12 Some History -- 3. Transversality Theorems -- 3.1 Introduction -- 3.2 Countability Properties in Topology -- 3.3 Negligible Subsets -- 3.4 The Complement of the Image of a Submanifold -- 3.5 Sard’s Theorem -- 3.6 Critical Points, Submersions and the Geometrical Form of Sard’s Theorem -- 3.7 The Transversality Theorem: Weak Form -- 3.8 Jet Spaces -- 3.9 The Thorn Transversality Theorem -- 3.10 Some History -- 4. Classification of Differentiable Functions -- 4.1 Introduction -- 4.2 Taylor Formulae Without Remainder -- 4.3 The Problem of Classification of Maps -- 4.4 Critical Points: the Hessian Form -- 4.5 The Morse Lemma -- 4.6 Bifurcations of Critical Points -- 4.7 Apparent Contour of a Surface in R3 -- 4.8 Maps from R2 into R2 -- 4.9 Envelopes of Plane Curves -- 4.10 Caustics -- 4.11 Genericity and Stability -- 5. Catastrophe Theory -- 5.1 Introduction -- 5.2 The Language of Germs -- 5.3 r-sufficient Jets; r-determined Germs -- 5.4 The Jacobian Ideal -- 5.5 The Theorem on Sufficiency of Jets -- 5.6 Deformations of a Singularity -- 5.7 The Principles of Catastrophe Theory -- 5.8 Catastrophes of Cusp Type -- 5.9 A Cusp Example -- 5.10 Liquid-Vapour Equilibrium -- 5.11 The Elementary Catastrophes -- 5.12 Catastrophes and Controversies -- 6. Vector Fields -- 6.1 Introduction -- 6.2 Examples of Vector Fields (Rn Case) -- 6.3 First Integrals -- 6.4 Vector Fields on Submanifolds -- 6.5 The Uniqueness Theorem and Maximal Integral Curves -- 6.6 Vector Fields and Line Fields. Elimination of the Time -- 6.7 One-parameter Groups of Diffeomorphisms -- 6.8 The Existence Theorem (Local Case) -- 6.9 The Existence Theorem (Global Case) -- 6.10 The Integral Flow of a Vector Field -- 6.11 The Main Features of a Phase Portrait -- 6.12 Discrete Flows and Continuous Flows -- 7. Linear Vector Fields -- 7.1 Introduction -- 7.2 The Spectrum of an Endomorphism -- 7.3 Space Decomposition Corresponding to Partition of the Spectrum -- 7.4 Norm and Eigenvalues -- 7.5 Contracting, Expanding and Hyperbolic Endomorphisms -- 7.6 The Exponential of an Endomorphism -- 7.7 One-parameter Groups of Linear Transformations -- 7.8 The Image of the Exponential -- 7.9 Contracting, Expanding and Hyperbolic Exponential Flows -- 7.10 Topological Classification of Linear Vector Fields -- 7.11 Topological Classification of Automorphisms -- 7.12 Classification of Linear Flows in Dimension 2 -- 8. Singular Points of Vector Fields -- 8.1 Introduction -- 8.2 The Classification Problem -- 8.3 Linearization of a Vector Field in the Neighbourhood of a Singular Point -- 8.4 Difficulties with Linearization -- 8.5 Singularities with Attracting Linearization -- 8.6 Lyapunov Theory -- 8.7 The Theorems of Grobman and Hartman -- 8.8 Stable and Unstable Manifolds of a Hyperbolic Singularity -- 8.9 Differentable Linearization: Statement of the Problem -- 8.10 Differentiable Linearization: Resonances -- 8.11 Differentiable Linearization: the Theorems of Sternberg and Hartman -- 8.12 Linearization in Dimension 2 -- 8.13 Some Historical Landmarks -- 9. Closed Orbits—Structural Stability -- 9.1 Introduction -- 9.2 The Poincaré Map -- 9.3 Characteristic Multipliers of a Closed Orbit -- 9.4 Attracting Closed Orbits -- 9.5 Classification of Closed Orbits and Classification of Diffeomorphisms -- 9.6 Hyperbolic Closed Orbits -- 9.7 Local Structural Stability -- 9.8 The Kupka-Smale Theorem -- 9.9 Morse-Smale Fields -- 9.10 Structural Stability Through the Ages -- 10.Bifurcations of Phase Portraits -- 10.1 Introduction -- 10.2 What Do We Mean by Bifurcation? -- 10.3 The Centre Manifold Theorem -- 10.4 The Saddle-Node Bifurcation -- 10.5 The Hopf Bifurcation -- 10.6 Local Bifurcations of a Closed Orbit -- 10.7 Saddle-node Bifurcation for a Closed Orbit -- 10.8 Period-doubling Bifurcation -- 10.9 Hopf Bifurcation for a Closed Orbit -- 10.10 An Example of a Codimension 2 Bifurcation -- 10.11 An Example of Non-local Bifurcation -- References -- Notation.Based on a lecture course at the Ecole Polytechnique (Paris), this text gives a rigorous introduction to many of the key ideas in nonlinear analysis, dynamical systems and bifurcation theory including catastrophe theory. Wherever appropriate it emphasizes a geometrical or coordinate-free approach which allows a clear focus on the essential mathematical structures. Taking a unified view, it brings out features common to different branches of the subject while giving ample references for more advanced or technical developments.Mathematics.Dynamics.Ergodic theory.Global analysis (Mathematics).Manifolds (Mathematics).Differential geometry.Mathematics.Differential Geometry.Global Analysis and Analysis on Manifolds.Dynamical Systems and Ergodic Theory.Springer eBookshttp://dx.doi.org/10.1007/978-3-642-57134-3URN:ISBN:9783642571343 |