Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T00:52:10.070Z Has data issue: false hasContentIssue false
  • English
  • Français

Fiber Completions, Contact Singularities and Single Valued Solutions for C∞-Second Order Ode

Published online by Cambridge University Press:  20 November 2018

Marek Kossowski
Marek Kossowski*
Affiliation:
Department of Mathematics University of South Carolina Columbia, SC 29208 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An implicitly defined second order ODE is said to be singular if the second derivative cannot be smoothly written in terms of lower order variables. The standard existence and uniqueness theory cannot be applied to such ODE and the graphs of solutions may fail to be regular curves (i.e., the solutions may have isolated C0-points or may fail to be single valued). In this paper we describe a local analysis for a large class of implicit second order ODE whose singular points satisfy a regularity condition. Within this class of ODE there is a secondary notion of (contact) singularity which is analogous to rest points for regular ODE. Theorems 5, 6, 7 and 8 produce invariants for these singularities which control the existence, uniqueness and the level of regularity in solutions.

Keywords

35A3035B3235F20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 4 , 01 August 1996 , pp. 849 - 870
Copyright
Copyright © Canadian Mathematical Society 1996

References

[A] Anosov, D.S., Arnold, V., Encyclopaedia Math. Sci., Springer, Verlag 1(1982).Google Scholar
[C] Cartan, E., Exterior Differential Systems and their Geometric Applications, Herman (1971).Google Scholar
[GG] Golubitsky, M., Guillemin, V., Stable Mappings and their singularities, Graduate Texts in, Math. 15(1975).Google Scholar
[KS] Kamran, N., Shadwick, J., Differential, J., Geom. 22(1985).Google Scholar
[Kl] Kossowski, M., First order PDE with singular solutions, Indiana Univ. Math., J. 35(1986).Google Scholar
[K2] Kossowski, M., Special points on first order PDE, Trans. Amer. Math., Soc. 1(1987).Google Scholar
[K3] Kossowski, M., Local Existence of Multivalued Solutions to Analytic Sympletic Monge Ampere Equations, Indiana Univ. Math., J. 40(1982).Google Scholar
[K4j Kossowski, M., Local Existence and Stability of Multivalued Solutions to Determined Analytic First Order Systems on the Plane, Duke Math., J. 69(1993).Google Scholar
[KKS] Kossowski, M., Kriele, M., Sluis, W., Fiber Completions, Contact Singularities and Single- Valued Solutions to C∞-Systems of First Order ODE, Nonlinearity, to appear.Google Scholar
[S] Sternberg, S., On the structure of local homeomorphisms of Euclidean n-spacell, Amer. J., Math. 80(1958).Google Scholar