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Numerical computation of symmetry-breaking bifurcation points

Published online by Cambridge University Press:  17 February 2009

B. S. Attili
B. S. Attili
Affiliation:
K.F.U.P.M., P.O. Box 1927, Dhahran 31261, Saudi Arabia
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Abstract

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We consider symmetry-breaking bifurcation points which arise in parameter-dependent nonlinear equations of the form f(x, λ) = 0. These types of bifurcation points are connected to pitchfork bifurcation points. A direct method is used to compute such points. Multiple shooting is used to discretise the two-point boundary-value problems to obtain a finite-dimensional problem.

Type
Research Article
Information
The ANZIAM Journal , Volume 35 , Issue 1 , July 1993 , pp. 103 - 113
Copyright
Copyright © Australian Mathematical Society 1993

References

[1] Attili, B., “Multiple shooting and the calculation of some types of singularities in B.V.P's”, Int. J. Comp. Math. 32 (1990) 97–111.CrossRefGoogle Scholar
[2] Attili, B., “A direct method for the characterization and computation of bifurcation points with corank 2”, Computing 48 (1992) 149–159.CrossRefGoogle Scholar
[3] Brezzi, F., Rappaz, J. and Raviart, P., “Finite dimensional approximation of nonlinear problems, Part III, Simple bifurcation points”, Numer. Math. 38 (1981) 1–30.CrossRefGoogle Scholar
[4] Griewank, A. and Reddien, G., “The approximation of generalized turning points by projection methods with super convergence to the critical parameter”, Num. Math. 14 (1975) 354–366.Google Scholar
[5] Griewank, A. and Reddien, G., “Characterization and computation of generalized turning points”, SIAM J. Numer. Anal. 21 (1984) 186–196.CrossRefGoogle Scholar
[6] Moore, G., “The numerical treatment of nontrivial bifurcation points”, Num. Funct. Anal. Optim. 2 (1980) 441–472.CrossRefGoogle Scholar
[7] Sattinger, D., “Group theoretic methods in bifurcation theory”, in Lecture Notes in Math. 762,(1979).CrossRefGoogle Scholar
[8] Seydel, R., “Branch switching in bifurcation problems for ordinary differential equation”, Num. Math. 41 (1983) 93–116.CrossRefGoogle Scholar
[9] Werner, B. and Spence, A., “The computation of symmetry breaking bifurcation points”, SIAM J. Num. Anal. 21 (1984) 388–399.CrossRefGoogle Scholar
[10] Werner, H., “On the numerical approximation of secondary bifurcation problems”, in Lecture Notes in Math. 878, (1981), 407–425.Google Scholar