An Introduction to Catastrophe Theory and Its Applications

This article is divided into two parts. In the first we give a description of the basic theorems of elementary catastrophe theory, along with heuristic explanations of why these theorems are valid. In particular, the main ideas in Mather’s proof of Thorn’s classification theorem are presented.The second part contains three applications of catastrophe theory to the buckling of beams, optics, and convex conservation laws. In these sections we attempt to state the problems precisely, to show how catastrophe theory may be used in a mathematically rigorous fashion, and to state what new information can be obtained by the use of catastrophe theory.

[1]  Christopher Zeeman.  The classification of elementary catastrophes of codimension ≤ 5 , 1976 .

[2]  Yung-chen Lu.  Singularity Theory and an Introduction to Catastrophe Theory , 1980 .

[3]  J. Callahan.  Singularities and Plane Maps , 1974 .

[4]  D. Schaeffer.  A regularity theorem for conservation laws , 1973 .

[5]  Kyoji Saito.  Einfach-elliptische Singularitäten , 1974 .

[6]  Theodor Bröcker,et al.  Differentiable Germs and Catastrophes , 1975 .

[7]  G B Kolata,et al.  Catastrophe theory: the emperor has no clothes. , 1977, Science.

[8]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[9]  Ian Stewart,et al.  Taylor expansions and catastrophes , 1976 .

[10]  M. Golubitsky,et al.  Stability of shock waves for a single conservation law , 1975 .

[11]  Wolfgang Meyer,et al.  On differentiable functions with isolated critical points , 1969 .

[12]  E. Bierstone.  Generic Equivariant Maps , 1977 .

[13]  V. Arnold.  LECTURES ON BIFURCATIONS IN VERSAL FAMILIES , 1972 .

[14]  James J. Callahan,et al.  Singularities and Plane Maps II: Sketching Catastrophes , 1977 .

[15]  R. Thom.  Stabilité structurelle et morphogenèse , 1974 .

[16]  G. N. Tjurina.  THE TOPOLOGICAL PROPERTIES OF ISOLATED SINGULARITIES OF COMPLEX SPACES OF CODIMENSION ONE , 1968 .

[17]  S. E. Khaikin,et al.  Theory of Oscillators , 1966 .

[18]  E. C. Zachmanoglou,et al.  Introduction to partial differential equations with applications , 1976 .

[19]  Peter D. Lax,et al.  THE FORMATION AND DECAY OF SHOCK WAVES , 1972 .

[20]  Gordon Wassermann,et al.  Stability of unfoldings in space and time , 1975 .

[21]  J. Mather,et al.  Stability of C∞ mappings, I, the division theorem , 1968 .

[22]  Giles W Hunt,et al.  A general theory of elastic stability , 1973 .

[23]  Johannes J. Duistermaat,et al.  Oscillatory integrals, lagrange immersions and unfolding of singularities , 1974 .

[24]  Gordon S Wassermann,et al.  Stability of Unfoldings , 1974 .

[25]  J. Moser.  On the volume elements on a manifold , 1965 .

[26]  J. Guckenheimer.  Solving a single conservation law , 1975 .