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

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

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.

Citation: J.W. Bruce, F. Tari. Generic 1-parameter families of binary differential equations of Morse type. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 79-90. doi: 10.3934/dcds.1997.3.79
[1]

Farid Tari. Two parameter families of binary differential equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 759-789. doi: 10.3934/dcds.2008.22.759

[2]

Farid Tari. Two-parameter families of implicit differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 139-162. doi: 10.3934/dcds.2005.13.139

[3]

Miriam Manoel, Patrícia Tempesta. Binary differential equations with symmetries. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1957-1974. doi: 10.3934/dcds.2019082

[4]

Jijiang Sun, Shiwang Ma. Nontrivial solutions for Kirchhoff type equations via Morse theory. Communications on Pure & Applied Analysis, 2014, 13 (2) : 483-494. doi: 10.3934/cpaa.2014.13.483

[5]

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025

[6]

Kai-Uwe Schmidt, Jonathan Jedwab, Matthew G. Parker. Two binary sequence families with large merit factor. Advances in Mathematics of Communications, 2009, 3 (2) : 135-156. doi: 10.3934/amc.2009.3.135

[7]

Ralf W. Wittenberg. Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5325-5357. doi: 10.3934/dcds.2014.34.5325

[8]

Suqi Ma, Zhaosheng Feng, Qishao Lu. A two-parameter geometrical criteria for delay differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 397-413. doi: 10.3934/dcdsb.2008.9.397

[9]

Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283

[10]

Ferenc Hartung. Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611

[11]

Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks & Heterogeneous Media, 2016, 11 (3) : 527-543. doi: 10.3934/nhm.2016008

[12]

Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475

[13]

Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877

[14]

Christian Bonatti, Sylvain Crovisier and Amie Wilkinson. The centralizer of a $C^1$-generic diffeomorphism is trivial. Electronic Research Announcements, 2008, 15: 33-43. doi: 10.3934/era.2008.15.33

[15]

Eduardo Liz, Manuel Pinto, Gonzalo Robledo, Sergei Trofimchuk, Victor Tkachenko. Wright type delay differential equations with negative Schwarzian. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 309-321. doi: 10.3934/dcds.2003.9.309

[16]

Tetsutaro Shibata. Boundary layer and variational eigencurve in two-parameter single pendulum type equations. Communications on Pure & Applied Analysis, 2006, 5 (1) : 147-154. doi: 10.3934/cpaa.2006.5.147

[17]

Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048

[18]

Hedia Fgaier, Hermann J. Eberl. Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation. Conference Publications, 2009, 2009 (Special) : 230-239. doi: 10.3934/proc.2009.2009.230

[19]

Thomas Bartsch, Anna Maria Micheletti, Angela Pistoia. The Morse property for functions of Kirchhoff-Routh path type. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1867-1877. doi: 10.3934/dcdss.2019123

[20]

Christian Bonatti, Sylvain Crovisier, Amie Wilkinson. $C^1$-generic conservative diffeomorphisms have trivial centralizer. Journal of Modern Dynamics, 2008, 2 (2) : 359-373. doi: 10.3934/jmd.2008.2.359

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]