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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.
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Abstract

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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

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