Abstract
At first glance, the most natural classification principle is classifying by codimension beginning with small codimensions. To classify objects “up to codimension ⩽ k” means to represent the entire space of objects studied as a finite union of submanifolds of codimensions not greater than k (called classes) and a remainder of codimension ⩾k + 1 so that within each class an object’s properties that are of interest to us do not change. Then all objects in typical, no more than k-parameter families, belong to our classes: the remaining ones may be avoided by a small perturbation of the family.
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Notes
The coefficient 11/ 10, forgotten in Khesin’s article, is necessary. Normal forms of the gradient families correspond for example to the values \( p = \pm 1{\text{ and}} - 1/2{\text{for }}D_4^ + \), and to the value \( p = 1{\text{ for }}D_4^ - \).
The investigations of this situation in catastrophe theory (R. Thorn (1972), L. Dara (1975) and F. Takens (1976)) basically repeat the work of M. Cibrario (1932), and A.A. Shestakov, and A.V. Pkhakadze (1959) carried out decades earlier.
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© 1994 Springer-Verlag Berlin Heidelberg
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Arnol’d, V.I. (1994). Classifications of Singularities and Catastrophes. In: Arnol’d, V.I. (eds) Dynamical Systems V. Encyclopaedia of Mathematical Sciences, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57884-7_11
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DOI: https://doi.org/10.1007/978-3-642-57884-7_11
Publisher Name: Springer, Berlin, Heidelberg
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