Catastrophe Theory

Singularity 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 "e;Singularities of ray systems"e; and "e;Singularities in the calculus of variations"e; listed in the bi- bliography of the present edition. Moscow, September 1983 v. I. Arnold Preface to the Russian Edition "e;Experts discuss forecasting disasters"e; said a New York Times report on catastrophe theory in November 1977. The London Times declared Catastrophe Theory to be the "e;main intellectual movement of the century"e; while an article on catastrophe theory in Science was headed "e;The emperor has no clothes"e;. 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 .