Numerical detection of symmetry breaking bifurcation points with nonlinear degeneracies
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- by Klaus Böhmer, Willy Govaerts and Vladimír Janovský PDF
- Math. Comp. 68 (1999), 1097-1108 Request permission
Abstract:
A numerical tool for the detection of degenerated symmetry breaking bifurcation points is presented. The degeneracies are classified and numerically processed on $1$-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.References
- 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.
- 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, DOI 10.1007/BF02238535
- 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, DOI 10.1016/0377-0427(89)90150-7
- K. Gatermann and R. Lauterbach, Automatic classification of normal forms, Preprint SC 95-3 (Februar 1995), Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany
- 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, DOI 10.1007/978-1-4612-5034-0
- 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, DOI 10.1007/978-1-4612-4574-2
- W. Govaerts, Computation of singularities in large nonlinear systems, SIAM J. Num. Anal. 34 (1997) pp. 867–880.
- V. Janovský and P. Plecháč, Numerical applications of equivariant reduction techniques, in Bifurcation and Symmetry, M. Golubitsky. E. Allgower, K. Böhmer, ed., Birkhäser Verlag, 1992.
- 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, DOI 10.1137/0520004
- 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
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
- Vladimír Janovský
- Affiliation: Faculty of Mathematics and Physics, Charles University, Prague, Czech Republik
- Email: janovsky@ms.mff.cuni.cz
- 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 - © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 1097-1108
- MSC (1991): Primary 65H10, 58C27, 47H15, 20C30
- DOI: https://doi.org/10.1090/S0025-5718-99-01052-2
- MathSciNet review: 1627846