Abstract
We show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations. We extend the approach by Werner (1988) by presenting methods for the detection of bifurcation points and the computation of (multiple) Hopf points. The essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group Γ. The group theoretical tool is an a priori knowledge of the interaction of certain subgroups Σ0 and Σ of Γ at (possibly multiple) steady state or Hopf bifurcation points (minimal Σ0-Σ-breaking bifurcation). We introduce a bifurcation graph which shows graphically this a priori information — its edges represent possible symmetry breaking bifurcations. Our analysis follows the lines of Golubitsky, Stewart and Schaeffer (1988) but it is aimed to numerical applications. We have chosen a 4-box-Brusselator model in order to explain our notions and ideas and to discuss the numerical procedure.
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References
A. Bossavit, Symmetry, groups and boundary value problems. A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry, Comput. Methods Appl. Mech. Engrg. 56 (1986) 167–215.
F. Brezzi, J. Rappaz and P.A. Raviart, Finite dimensional approximations of nonlinear problems, Part III: Simple Bifurcation points, Numer. Math. 38 (1981) 1–30.
E. Buzano, G. Geymonat and T. Poston, Post-buckling behaviour of a non-linearly hyperelastic thin rod with cross-section invariant under the dihedral group Dn, Archional Rational Mech. Anal. 89 (1985) 307–388.
G. Cicogna, Symmetry breakdown from bifurcation, Lett. Nuovo Cimento 31 (1981) 600–602.
B. De Dier, V. Hlavacek and P. Van Rompay, Analysis of dissipative structures in a two-dimensional autocatalytic system: the Brusselator, Report, Universiteit Leuven, 1986.
P. Deuflhard, B. Fiedler and P. Kunkel, Efficient numerical pathfollowing beyond critical points, SIAM J. Numer. Anal. 24 (1987) 912–927.
M. Field and R.W. Richardson, Symmetry breaking and the maximal isotropy subgroup conjecture for finite reflection groups, Arch. Rational Mech. Anal., to appear.
H. Fujii and M. Yamaguti, Structure of singularities and its numerical realization in nonlinear elasticity, J. Math. Kyoto Univ. 20(3) (1980) 489–590.
M. Golubitsky, The Benard problem, symmetry and the lattice of isotropy subgroups, in: C.P. Borter et al., Eds., Bifurcation Theory, Mechanics & Physics (Reidel, Dordrecht, 1983) 225–256.
M. Golubitsky and I. Stewart, Hopf bifurcation in the presence of symmetry, Arch. Rational Mech. Anal. (1985) 107-165.
M. Golubitsky and I. Stewart, Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators, in: M. Golubitsky and J. Guckenheimer, Eds., Multiparameter Bifurcation Theory, Contemporary Mathematics 56 (Amer. Math. Soc, Providence, 1986) 131–173.
M. Golubitsky, I. Stewart and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 2 (Springer, Berlin, 1988).
A. Griewank and G. Reddien, The calculations of Hopf points by a direct method, IMA J. Numer. Anal. 3 (1983) 295–303.
T. Healey, Global bifurcation and continuation in the presence of symmetry with an application to solid mechanics, Cornell University preprint, 1986.
T. Healy, A group-theoretic approach to computational bifurcation with symmetry, Comput. Methods Appl. Mech. Engrg. 67 (1988), 257–295.
H. Jarausch and W. Mackens, Computing bifurcation diagrams for large nonlinear variational problems, in: Deuflhard and Engquist, Eds., Progress in Large Scale Scientific Computing (Birkhäuser, Basel, 1987).
A.D. Jepson, Numerical Hopf bifurcation, Ph.D. Thesis, Part II, California Institute of Technology, Pasadena, CA, 1981.
M. Kubicek and M. Marek, Computational Methods in Bifurcation Theory and Dissipative Structures (Springer, Berlin, 1983).
I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. II, J. Chem. Physics 48 (1968) 1695–1700.
D.S. Riley and K.H. Winters, The onset of convection in a porous medium: a preliminary study, United Kingdom Atomic Energy Authority Harwell, Oxfordshire, 1987.
D. Roose and V. Hlavacek, A direct method for the computation of Hopf bifurcation points, SIAM J. Appl. Math. 45 (1985) 879–894.
R. Seydel, Efficient branch switching in systems of nonlinear equations, in: J. Albrecht et al., Eds., Numerical Treatment of Eigenvalue Problems, Vol. 3, Internat. Ser. Numer. Math. 69 (Birkhäuser, Basel, 1983) 181–191.
A. Steindl and H. Troger, Bifurcation of the equilibrium of a spherical double pendulum at a multiple eigenvalue, in: Internat. Ser. Numer. Math. 79 (Birkhäuser, Basel, 1987) 277–287.
A. Vanderbauwhede, Local Bifurcation Theory and Symmetry (Pitman, London, 1982).
B. Werner, Regular systems for bifurcation points with underlying symmetries, in: T. Küpper and H.D. Mittelmann and H. Weber, Eds., Numerical Methods for Bifurcation Problems, Internat. Ser. Numer. Math. 70 (Birkhäuser, Basel, 1984) 562–574.
B. Werner, Computational methods for bifurcation problems with symmetries and applications to steady states of n-box reaction-diffusion models, in: D.F. Griffith and G.A. Watson, Eds., 1987 Dundee Conference on Numerical Analysis (Pitman, London, 1988).
B. Werner and A. Spence, The computation of symmetry-breaking bifurcation points, SIAM J. Numer. Anal. 21 (1984) 388–399.
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Dellnitz, M., Werner, B. (1990). Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points. In: Mittelmann, H.D., Roose, D. (eds) Continuation Techniques and Bifurcation Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 92. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5681-2_7
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DOI: https://doi.org/10.1007/978-3-0348-5681-2_7
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