Catastrophe TheorySingularity theory is growing very fast and many new results have been discovered since the Russian edition appeared: for instance the relation of the icosahedron to the problem of by passing a generic obstacle. The reader can find more details about this in the articles "Singularities of ray systems" and "Singularities in the calculus of variations" listed in the bi bliography of the present edition. Moscow, September 1983 v. I. Arnold Preface to the Russian Edition "Experts discuss forecasting disasters" said a New York Times report on catastrophe theory in November 1977. The London Times declared Catastrophe Theory to be the "main intellectual movement of the century" while an article on catastrophe theory in Science was headed "The emperor has no clothes". This booklet explains what catastrophe theory is about and why it arouses such controversy. It also contains non-con troversial results from the mathematical theories of singulari ties and bifurcation. The author has tried to explain the essence of the fundamen tal results and applications to readers having minimal mathe matical background but the reader is assumed to have an in quiring mind. Moscow 1981 v. I. Arnold Contents Chapter 1. Singularities, Bifurcations, and Catastrophe Theories ............... 1 Chapter 2. Whitney's Singularity Theory ... 3 Chapter 3. Applications of Whitney's Theory 7 Chapter 4. A Catastrophe Machine ...... 10 Chapter 5. Bifurcations of Equilibrium States 14 Chapter 6. Loss of Stability of Equilibrium and the Generation of Auto-Oscillations . . . . . . 20 . |
Contents
Singularities Bifurcations and Catastrophe | 1 |
Applications of Whitneys Theory | 7 |
Bifurcations of Equilibrium States | 14 |
Loss of Stability of Equilibrium and | 20 |
Singularities of Stability Boundaries and | 27 |
Large Scale Distribution of Matter in | 39 |
Singularities of Accessibility Boundaries | 47 |
Smooth Surfaces and Their Projections | 57 |
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accessibility boundary accessibility domain applications attractor axis bifurcation bundle called catas catastrophe theory co-ordinates complex consider Contact Geometries contact structure control parameters convex hull curvature cusp ridge cusp singularities density described dimension disk dual equations Euclidean space example field of limit folds and cusps Gaussian mapping geodesic geometry gradient mapping gularities horizontal indicatrix inflection instance intersection jump larities limit curves limit directions linear loss of stability mathematics maxima function medium mensional metamorphoses neighbourhood normal forms normal mapping obstacle surface order tangents oscillation param parameter family particles phase curves phase space plane problem projection propagation rays René Thom segments singu singular point singularities of caustics singularity theory skew-scalar product small perturbation smooth function smooth mapping smooth surface submanifold subspace symplectic geometry symplectic manifold symplectic space symplectic structure target theorem Thom three dimensional space three-dimensional tions torus transformed trophe vectors velocity field visible contours wave front Whitney's theory