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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

A numerical Liapunov-Schmidt method with applications to Hopf bifurcation on a square


Authors: Peter Ashwin, Klaus Böhmer and Zhen Mei
Journal: Math. Comp. 64 (1995), 649-670, S19
MSC: Primary 65J15; Secondary 47H15, 47N20, 65N99
MathSciNet review: 1284661
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Abstract: We discuss an iterative method for calculating the reduced bifurcation equation of the Liapunov-Schmidt method and its numerical approximation. Using appropriate genericity assumptions (with symmetry), we derive a Taylor series for the reduced equation, where the bifurcation behavior is determined by its numerical approximation at a finite order of truncation. This method is used to calculate reduced equations at Hopf bifurcation of the two-dimensional Brusselator equations on a square with Neumann and Dirichlet boundary conditions. We examine several Hopf bifurcations within the three-parameter space. There are regions where we observe direct bifurcation to branches of periodic solutions with submaximal symmetry.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1995-1284661-X
PII: S 0025-5718(1995)1284661-X
Keywords: Hopf bifurcation, steady state bifurcation, Liapunov-Schmidt method, finite determinacy, equivariant operator
Article copyright: © Copyright 1995 American Mathematical Society