Hostname: page-component-7bb8b95d7b-qxsvm Total loading time: 0 Render date: 2024-09-12T23:22:22.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Completely integrable holonomic systems of first-order differential equations

Published online by Cambridge University Press:  14 November 2011

Shyuichi Izumiya
Shyuichi Izumiya
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

We classify completely integrable holonomic systems of first-order differential equations for one real-valued function by equivalence under the group of point transformations in the sense of Sophus Lie. In order to pursue the classification, we use the notion of one parameter Legendrian unfoldings which induces a special class of divergent diagrams of map germs which are called integral diagrams. Our normal forms are represented by integral diagrams.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 3 , 1995 , pp. 567 - 586
Copyright
Copyright © Royal Society of Edinburgh 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Arnol'd, V. I., Gusein-Zade, S. M. and Varchenko, A. N.. Singularities of Differentiable Maps, Vol. I (Boston: Birkhäser, 1986).Google Scholar
2Damon, J.. The Unfolding and Determinancy Theorems for Subgroups of A and K. Mem. Amer. Math. Soc. 50 (Providence, RI: American Mathematical Society, 1984).Google Scholar
3Dufour, J. P.. Families de courbes planes différentiables. Topology 22 (1983), 449–74.CrossRefGoogle Scholar
4Goryunov, V. V.. Geometry of bifurcation diagrams of simple projections onto the line. Functional Anal. Appl. 15 (1981), 77–82.CrossRefGoogle Scholar
5Goryunov, V. V.. Projection and vector fields, tangent to the discriminant of a complete intersection. Functional Anal. Appl. 22 (1988), 104–13.CrossRefGoogle Scholar
6Hayakawa, A., Ishikawa, G., Izumiya, S. and Yamaguchi, K.. Classification of generic integral diagram and first order ordinary differential equations. Int. J. Math. 5 (1994), 447–89.CrossRefGoogle Scholar
7Izumiya, S.. Generic bifurcations of varieties. Manuscripta Math. 46 (1984), 137–64.CrossRefGoogle Scholar
8Izumiya, S.. The theory of Legendrian unfoldings and first-order differential equations. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 517–32.CrossRefGoogle Scholar
9Zakalyukin, V. M.. Lagrangian and Legendrian singularities. Functional Anal. Appl. 10 (1976), 23–31.CrossRefGoogle Scholar
10Zakalyukin, V. M.. Reconstructions of fronts and caustics depending on a parameter and versality of mappings. J. Soviet Math. 27 (1983), 2713–35.CrossRefGoogle Scholar