Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T21:54:42.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Automating Symmetry-Breaking Calculations

Published online by Cambridge University Press:  01 February 2010

P. C. Matthews
P. C. Matthews
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom, paul.matthews@nottingham.ac.uk, http://www.maths.nottingham.ac.uk/personal/pcm

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The process of classifying possible symmetry-breaking bifurcations requires a computation involving the subgroups and irreducible representations of the original symmetry group. It is shown how this calculation can be automated using a group theory package such as GAP. This enables a number of new results to be obtained for larger symmetry groups, where manual computation is impractical. Examples of symmetric and alternating groups are given, and the method is also applied to the spatial symmetry-breaking of periodic patterns observed in experiments.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 7 , 2004 , pp. 101 - 119
Copyright
Copyright © London Mathematical Society 2004

References

Referances

1. Arbell, H. and Fineberg, J., ‘Spatial and temporal dynamics of two interacting modes in parametrically driven surface waves’, Phys. Rev. Lett. 81 (1998) 4384–4387.CrossRefGoogle Scholar
2. Ashwin, P., Podvigina, O., ‘Hopf bifurcation with cubic symmetry and instability of ABC flow’, Proc. Roy. Soc. A 459 (2003) 1801–1827.CrossRefGoogle Scholar
3. Bodenschatz, E., Pesch, W., Ahlers, G., ‘Recent developments in Rayleigh Bénard convection’, Ann. Rev. Fluid. Mech. 32 (2000) 709–778.CrossRefGoogle Scholar
4. Chossat, P., Lauterbach, R., Methods in equivariant bifurcations and dynamical systems, Adv. Ser. Nonlin. Dynam. 15 (World Scientific, River Edge, NJ, 2000).CrossRefGoogle Scholar
5. Cicogna, G., ‘Symmetry breakdown from bifurcations’, Lettere al Nuovo Cimento 31 (1981) 600–602.CrossRefGoogle Scholar
6. Cox, S. M., Matthews, P. C., ‘Instability and localisation of patterns due to a conserved quantity’, Physica D 175 (2003) 196–219.CrossRefGoogle Scholar
7. Dawes, J. H. P., Matthews, P. C., Rucklidge, A. M., ‘Reducible actions of D4⋉T2: superlattice patterns and hidden symmetries’, Nonlinearity 16 (2003) 615–645.CrossRefGoogle Scholar
8. The GAP Group, GAP - Groups, algorithms, and programming, Version 4.3 (2002); http://www.gap-system.org.Google Scholar
9. Golubitsky, M., Stewart, I., ‘An algebraic criterion for symmetric Hopf birfurcations’, Proc. Roy. Soc. London Ser. A 440 (1993) 727–732.Google Scholar
10. Golubitsky, M., Stewart, I., The symmetry perspective (Birkhäuser, Basel, 2002).CrossRefGoogle Scholar
11. Golubitsky, M., Stewart, I., Schaeffer, D. G., Singularities and groups in bifurcation theory, vol. II, Appl. Math. Ser. 69 (Springer, New York, 1988).Google Scholar
12. Hoyle, R. B., ‘Shapes and cycles arising at the steady bifurcation with icosahedral symmetry’, Physica D 191 (2004) 261–281.CrossRefGoogle Scholar
13. Ihrig, E., Golubitsky, M., ‘Pattern selection with 0(3) symmetry’, Physica D 13 (1984) 1–33.CrossRefGoogle Scholar
14. Kudrolli, A., Pier, B., Gollub, J. P., ‘Superlattice patterns in surface waves’ , Physica D, 123 (1998) 99–111.CrossRefGoogle Scholar
15. Lavassani, A.Lari, Langford, W. F., Huseyin, K., ‘Symmetry-breaking bifurcations on multidimensional fixed point subspaces’, Dynam. Stability Systems 9 (1994) 345–373.CrossRefGoogle Scholar
16. Lauterbach, R., ‘Spontaneous symmetry breaking in higher-dimensional fixed point spaces’, Z. Angew. Math. Phys. 43 (1992) 430–448.CrossRefGoogle Scholar
17. Linehan, M. J., Stedman, G. E., ‘Little groups of irreps of 0(3), SO(3), and the infinite axial subgroups’, J. Phys. A 34 (2001) 6663–6688.CrossRefGoogle Scholar
18. Matthews, P. C., ‘Bifurcation in two-dimensional fixed point subspaces’, preprint (2003); arXiv:math.DS/0305392.Google Scholar
19. Melbourne, I., ‘A singularity theory analysis of bifurcation problems with octahedral symmetry’, Dynamics and Stability of Systems 1 (1986) 293–321.CrossRefGoogle Scholar
20. Michel, M., ‘Symmetry defects and broken symmetry’, Rev. Modern Phys. 52(1980) 617–651.CrossRefGoogle Scholar
21. Rayleigh, Lord., ‘On convection currents in a horizontal layer of fluid, when the higher temperature is on the underside’, Philos. Mag. 32 (1916) 529–546.CrossRefGoogle Scholar
22. Sattinger, D. H., Group theoretic methods in bifurcation theory (Springer, New York, 1979).CrossRefGoogle Scholar
23. Swift, J. W., Barany, E., ‘Chaos in the Hopf bifurcation with tetrahedral symmetry: convection in a rotating fluid with low Prandtl number’, Proc, Nonlinear Hydro- dynamic Stability and- Transition, Sophia-Antipolis, (1990), European J. Mech. B Fluids 10 (1991) 99–104.Google Scholar
24. Tse, D. P., Rucklidge, A. M., Hoyle, R. B., Silber, M., ‘Spatial period-multiplying instabilities of hexagonal Faraday waves’, Physica D 146 (2000) 367–387.CrossRefGoogle Scholar
25. Vanderbauwhede, A., ‘'Symmetry and bifurcation near families of solution’, J. Differential Equations 36 (1980) 173–178.CrossRefGoogle Scholar