Skip to main content
main-content

Über dieses Buch

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 "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 .

Inhaltsverzeichnis

Frontmatter

Chapter 1. Singularities, Bifurcations, and Catastrophe Theories

Abstract
The first information on catastrophe theory appeared in the western press about ten years ago. In journals like “News Week” there were reports of a revolution in mathematics comparable with Newton’s invention of the differential and integral calculus. It was claimed that the new science, catastrophe theory, is much more valuable to mankind than mathematical analysis. While Newtonian theory only considers smooth, continuous processes, catastrophe theory provides a universal method for the study of all jump transitions, discontinuities, and sudden qualitative changes. There appeared hundreds of scientific and popular science publications in which catastrophe theory was applied to such diverse fields as, for instance, the study of heart beat, geometrical and physical optics, embryology, linguistics, experimental psychology, economics, hydrodynamics, geology, and the theory of elementary particles. Among the published works on catastrophe theory are studies of stability of ships, models for the activity of the brain and mental disorders, for rioting prisoners and for investors on the stock exchange, studies of the effects of alcohol on drivers and of the censor’s attitude towards pornographic literature.
Vladimir Igorevich Arnold

Chapter 2. Whitney’s Singularity Theory

Abstract
In 1955 the American mathematician Hassler Whitney published the article ‘Mappings of the plane into the plane’ laying the foundations for a new mathematical theory of singularities of smooth mappings.
Vladimir Igorevich Arnold

Chapter 3. Applications of Whitney’s Theory

Abstract
Since smooth mapping are found everywhere, their singularities must be everywhere also and since Whitney’s theory gives significant information on singularities of generic mappings, we can try to use this information to study large numbers of different phenomena and processes in all areas of science. This sample idea is the whole essence of catastrophe theory.
Vladimir Igorevich Arnold

Chapter 4. A Catastrophe Machine

Abstract
In contrast to the example given above the application of singularity theory to the study of bifurcation of equilibrium states in the theory of elasticity is irreproachably founded.
Vladimir Igorevich Arnold

Chapter 5. Bifurcations of Equilibrium States

Abstract
An evolutionary process is described mathematically by a vector field in phase space. A point of phase space is called a state of the system. The vector at this point indicates the speed of change of the state.
Vladimir Igorevich Arnold

Chapter 6. Loss of Stability of Equilibrium and the Generation of Auto-Oscillations

Abstract
Loss of stability of an equilibrium state on change of parameter is not necessarily associated with the bifurcation of this state. An equilibrium state can lose stability without even interacting with another state.
Vladimir Igorevich Arnold

Chapter 7. Singularities of Stability Boundaries and the Principle of the Fragility of Good Things

Abstract
We consider an equilibrium state of a system depending on several parameters and assume that (in some domain of variation of the parameters) this equilibrium state does not bifurcate.
Vladimir Igorevich Arnold

Chapter 8. Caustics, Wave Fronts and Their Metamorphoses

Abstract
One of the most important deductions of singularity theory is the universality of certain simple forms like folds and cusps which are encountered everywhere and which one should learn to recognise. As well as the singularities already described one often meets some further types called the ‘swallow tail’, ‘pyramid’, ‘purse’ and so on.
Vladimir Igorevich Arnold

Chapter 9. Large Scale Distribution of Matter in the Universe

Abstract
At the present time the distribution of matter in the universe is highly non-uniform (there are planets, the sun, stars, galaxies, clusters of galaxies and so on). Astrophysists nowadays suggest that in earlier stages of the development of the universe there was no such non-uniformity. How did it come about? Zel’dovich in 1970 proposed an explanation of the formation of particle clusters that is mathematically equivilent to the analysis the formation of singularities of caustics.
Vladimir Igorevich Arnold

Chapter 10. Singularities in Optimization Problems, the Maxima Function

Abstract
Many singularities, bifurcations, and catastrophes (jumps) arise in all maxima and minima problems ; such problem arise, for instance, in optimization, control theory and decision theory. For instance, suppose we have to find x such that the value of a function f (x ) is maximal (Fig. 46). On smooth change of the function the optimum solution changes with a jump, transferring from one competing maxima (A) to the other (B).
Vladimir Igorevich Arnold

Chapter 11. Singularities of Accessibility Boundaries

Abstract
A control system in phase space is defined as follows: at every point of the space we have not one velocity vector (as in the usual evolutionary system) but a whole set of vectors called the indicatrix of permissible velocities (Fig. 49).
Vladimir Igorevich Arnold

Chapter 12. Smooth Surfaces and Their Projections

Abstract
A smooth curve on the plane can have a tangent which touches it at any number of points (Fig. 56), but this is not true for generic curves since a small perturbation can always produce a curve such that no straight line is tangential at more than two points.
Vladimir Igorevich Arnold

Chapter 13. Problems of By-Passing Obstacles

Abstract
We consider an obstacle in three-dimensional space bounded by a smooth surface (Fig. 60). It is clear that the shortest path from x to y avoiding the obstacle consists of line segments and segments of geodesics (curves of minimal length) on the obstacle surface. The geometry of the shortest paths is greatly affected by the inflections of the obstacle surface.
Vladimir Igorevich Arnold

Chapter 14. Symplectic and Contact Geometries

Abstract
Many problems in singularity theory (for instance, the classifications of singularities of caustics and of wavefronts and also of determining the singularities in optimization and variational calculus problems) are understandable only in terms of the geometries of symplectic and contact manifolds. These geometries are pleasantly different from the three usual ones (those of Euclid, Lobachevskii and Riemann).
Vladimir Igorevich Arnold

Chapter 15. The Mystics of Catastrophe Theory

Abstract
The applications of singularity theory to the natural sciences are not the only aspects of catastrophe theory; along with the concrete investigations of the Zeeman type one has the more philosophical work of the mathematician René Thorn who first revealed the universality of the Whitney singularity theory (and the preceding work of Poincaré and Andronov on bifurcation theory), introduced the term ‘catastrophe theory’ and has been a great propagandist for the subject.
Vladimir Igorevich Arnold

Backmatter

Weitere Informationen