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AuthorDemazure, Michel. author
TitleBifurcations and Catastrophes [electronic resource] : Geometry of Solutions to Nonlinear Problems / by Michel Demazure
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2000
Connect tohttp://dx.doi.org/10.1007/978-3-642-57134-3
Descript VIII, 304 p. 1 illus. online resource

SUMMARY

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


CONTENT

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 Sardโ{128}{153}s Theorem -- 3.6 Critical Points, Submersions and the Geometrical Form of Sardโ{128}{153}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 Orbitsโ{128}{148}Structural Stability -- 9.1 Introduction -- 9.2 The Poincarรฉ 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


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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