Numerical detection

of symmetry breaking bifurcation points

with nonlinear degeneracies

Authors:
Klaus Böhmer, Willy Govaerts and Vladimí r Janovský

Journal:
Math. Comp. **68** (1999), 1097-1108

MSC (1991):
Primary 65H10, 58C27, 47H15, 20C30

Published electronically:
February 13, 1999

MathSciNet review:
1627846

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A numerical tool for the detection of degenerated symmetry breaking bifurcation points is presented. The degeneracies are classified and numerically processed on -D restrictions of the bifurcation equation. The test functions that characterise each of the equivalence classes are constructed by means of an equivariant numerical version of the Liapunov-Schmidt reduction. The classification supplies limited qualitative information concerning the imperfect bifurcation diagrams of the detected bifurcation points.

**1.**K. BÖHMER, W. GOVAERTS AND V. JANOVSKÝ,*Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies*, Bericht zur Fachbereich Mathematik der Philipps-Universität Marburg 1996.**2.**K. Böhmer,*On a numerical Liapunov-Schmidt method for operator equations*, Computing**51**(1993), no. 3-4, 237–269 (English, with English and German summaries). MR**1253405**, 10.1007/BF02238535**3.**Michael Dellnitz and Bodo Werner,*Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points*, J. Comput. Appl. Math.**26**(1989), no. 1-2, 97–123. Continuation techniques and bifurcation problems. MR**1007355**, 10.1016/0377-0427(89)90150-7**4.**K. GATERMANN AND R. LAUTERBACH,*Automatic classification of normal forms*, Preprint SC 95-3 (Februar 1995), Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany**5.**Martin Golubitsky and David G. Schaeffer,*Singularities and groups in bifurcation theory. Vol. I*, Applied Mathematical Sciences, vol. 51, Springer-Verlag, New York, 1985. MR**771477****6.**Martin Golubitsky, Ian Stewart, and David G. Schaeffer,*Singularities and groups in bifurcation theory. Vol. II*, Applied Mathematical Sciences, vol. 69, Springer-Verlag, New York, 1988. MR**950168****7.**W. GOVAERTS,*Computation of singularities in large nonlinear systems*, SIAM J. Num. Anal. 34 (1997) pp. 867-880. CMP**97:13****8.**V. JANOVSKÝ AND P. PLECHÁC,*Numerical applications of equivariant reduction techniques*, in Bifurcation and Symmetry, M. Golubitsky. E. Allgower, K. Böhmer, ed., Birkhäser Verlag, 1992. CMP**94:04****9.**A. D. Jepson and A. Spence,*On a reduction process for nonlinear equations*, SIAM J. Math. Anal.**20**(1989), no. 1, 39–56. MR**977487**, 10.1137/0520004**10.**Bodo Werner,*The numerical analysis of bifurcation problems with symmetries based on bordered Jacobians*, Exploiting symmetry in applied and numerical analysis (Fort Collins, CO, 1992) Lectures in Appl. Math., vol. 29, Amer. Math. Soc., Providence, RI, 1993, pp. 443–457. MR**1247745**

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Additional Information

**Klaus Böhmer**

Affiliation:
Philipps Universität, Fachbereich Mathematik, Marburg, Germany

Email:
boehmer@mathematik.uni-marburg.de

**Willy Govaerts**

Affiliation:
Department of Applied Mathematics and Computer Science, University of Genh, Belgium

Email:
Willy.Govaerts@rug.ac.be

**Vladimí r Janovský**

Affiliation:
Faculty of Mathematics and Physics, Charles University, Prague, Czech Republik

Email:
janovsky@ms.mff.cuni.cz

DOI:
http://dx.doi.org/10.1090/S0025-5718-99-01052-2

Keywords:
Symmetry-breaking bifurcation,
nonlinear degeneracy,
bordered matrices,
generalised Liapunov-Schmidt reduction,
Newton-like method,
pathfollowing

Received by editor(s):
February 16, 1996

Received by editor(s) in revised form:
August 8, 1997, and December 16, 1997

Published electronically:
February 13, 1999

Additional Notes:
The first author was partially supported by the Volkswagen Foundation and the Deutsche Forschungsgemeinschaft

The second author was partially supported by the Fund for Scientific Research F.W.O., Gent, Belgium

The third author was partially supported by the grants GAČR 201/98/0528 and GAUK 96/199

Article copyright:
© Copyright 1999
American Mathematical Society