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January  1997, 3(1): 79-90. doi: 10.3934/dcds.1997.3.79

Generic 1-parameter families of binary differential equations of Morse type

J.W. Bruce 1, and  F. Tari 1,

1. 

Department of Pure Mathematics, The University of Liverpool, P.O. Box 147, Liverpool L69 3BX, United Kingdom, United Kingdom

Received  January 1996 Revised  September 1996 Published  October 1996

In a previous paper [2] we made a classification of generic binary differential equations (BDE's)

$a(x,y)dy^2+2b(x,y)dxdy+c(x,y)dx^2=0$

near points at which the discriminant function $b^2-ac$ has a Morse singularity. Such points occur naturally in families of BDE's and here we describe the manner in which the configuration of solution curves change in their natural 1-parameter versal deformations.
The results in this paper can be used to describe, for instance, the changes in the structure of the asymptotic curves on a 1-parameter family of smooth surfaces acquiring a flat umbilic and on integral curves determined by eigenvectors of 1-parameter families of $2\times 2$ matrices. It also sheds light on the structure of the rarefraction curves associated to a $2\times 2$ system of conservation laws in 1 space variable.

Mathematics Subject Classification: 58Fxx, 34Cx.
Citation: J.W. Bruce, F. Tari. Generic 1-parameter families of binary differential equations of Morse type. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 79-90. doi: 10.3934/dcds.1997.3.79
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