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

J.W. Bruce, F. Tari

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

In a previous paper [2] we made a classification of generic binary differential equations (BDE's) near points at which the discriminant function 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 matrices. It also sheds light on the structure of the rarefraction curves associated to a system of conservation laws in 1 space variable. Mathematics Subject Classification: 58Fxx, 34Cxx.
Original languageEnglish
Pages (from-to)79-90
JournalDiscrete and Continuous Dynamical Systems - Series A (DCDS-A)
Volume3
Issue number1
DOIs
Publication statusPublished - 1997

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Binary
Differential equation
Curve
Versal Deformation
Discriminant Function
Systems of Conservation Laws
Smooth surface
Eigenvector
Singularity
Configuration
Family

Cite this

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title = "Generic 1-parameter families of binary differential equations of Morse type",
abstract = "In a previous paper [2] we made a classification of generic binary differential equations (BDE's) near points at which the discriminant function 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 matrices. It also sheds light on the structure of the rarefraction curves associated to a system of conservation laws in 1 space variable. Mathematics Subject Classification: 58Fxx, 34Cxx.",
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Generic 1-parameter families of binary differential equations of Morse type. / Bruce, J.W.; Tari, F.

In: Discrete and Continuous Dynamical Systems - Series A (DCDS-A), Vol. 3, No. 1, 1997, p. 79-90.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Generic 1-parameter families of binary differential equations of Morse type

AU - Bruce, J.W.

AU - Tari, F.

PY - 1997

Y1 - 1997

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AB - In a previous paper [2] we made a classification of generic binary differential equations (BDE's) near points at which the discriminant function 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 matrices. It also sheds light on the structure of the rarefraction curves associated to a system of conservation laws in 1 space variable. Mathematics Subject Classification: 58Fxx, 34Cxx.

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DO - 10.3934/dcds.1997.3.79

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