Abstract
In these lectures I would like to discuss how the existence of symmetries alters the type of bifurcation behavior that one expects to observe. In the first lecture I will concentrate on the structure and dynamics of steady-state bifurcation from equilibria. It is here that the influence of symmetries on linearized equations will be discussed and some facts from elementary representation theory introduced. The second lecture will be devoted to effects of symmetry on period-doubling in maps with a short description of an application to large arrays of Josephson junctions. In the final lecture I will describe how certain standard choices of boundary conditions (particularly Neumann) can be thought of as symmetry constraints and how this fact alters notions of genericity. It accord with the style that has developed in the lectures at this workshop, the lectures are of different length.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Armbruster & G. Dangelmayr, Coupled stationary bifurcations in non-flux boundary value problems, Math. Proc. Comb. Phil. Soc. 101 (1987), 167–192.
D. Armbruster, J. Guckenheimer & P. Holmes, Heteroclinic cycles and modulated travelling waves in systems with 0(2) symmetry, Physica 29 D (1988) 257–282.
D.G. Aronson, M. Golubitsky & M. Krupa, Coupled arrays of Josephson junctions and bifurcation of maps with S N symmetry, Nonlinearity. Submitted.
F.H. Busse & K.E. Heikes, Convection in a rotating layer: a simple case of turbulence, Science 208 (1980) 173–175.
P. Chossat & M. Golubitsky, Symmetry-increasing bifurcation of chaotic attractors, Physica 32 D (1988) 423–436.
G. Cicogna, Symmetry breakdown from bifurcations, Lettere al Nuovo Cimento 31 (1981) 600–602.
S. Ciliberto & J. Gollub, Chaotic mode competition in parametrically forced surface waves, J. Fluid Mech. 158 (1985), 381–398.
J.D. Crawford, M. Golubitsky, M.G.M. Gomes, E. Knobloch & I.N. Stewart, Boundary conditions as symmetry constraints, Preprint, University of Warwick (1989).
J.D. Crawford, E. Knobloch & H. Riecke, Competing parametric instabilities with circular symmetry, Phys. Lett. A 135 (1989) 20–24.
M. Field, Equivariant dynamical systems, Trans. A.M.S. 259, No. 1 (1980) 185–205.
M. Field, Equivariant bifurcation theory and symmetry breaking, Dyn. Diff. Eqn. 1, No. 4 (1989), 369–421.
M. Field & R.W. Richardson, Symmetry breaking and the maximal isotropy subgroup conjecture for reflection groups, Arch. Rational Mech. Anal. 105 (1989) 61–94.
M. Field & R.W. Richardson, New examples of symmetry breaking and the distribution of symmetry breaking isotropy types, In preparation.
H. Fujii, M. Mimura & Y. Nishiura, A picture of the global bifurcation diagram in ecologically interacting and diffusing systems, Physica 5D (1982) 1–42.
J. Guckenheimer & P. Holmes, Nonlinear Oscillations, Dynamical Systems, and bifurcation of Vector Fields,, Appl. Math. Sci. 42, Springer-Verlag, New York, 1983.
J. Guckenheimer & P. Holmes, Structurally stable heteroclinic cycles, Math. Proc. Comb. Phil. Soc. 103, part 1 (1988), 189–192.
M. Golubitsky, I.N. Stewart & D.G. Schaeffer, Singularities and Groups in Bifurcation Theory: Vol. II, Applied Math. Sci. 69, Springer-Verlag, New York, 1988.
P. Hadley, M.R. Beasley & K. Wiesenfeld, Phase locking of Josephson-junction series arrays, Phys. Rev. B 38, No. 13 (1988) 8712–8719.
P. Hadley, M.R. Beasley & K. Wiesenfeld, Phase locking of J osephson-junction arrays, Applied Phys. Lett. 52, No. 19 (1988), 1619–1621.
E. Ihrig & M. Golubitsky, Pattern section with 0(3) symmetry, Physica 12D (1984), 1–33.
R.M. May & W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975), 243–253.
I. Melbourne, Intermittancy as a codimension three phenomenon, Dyn. Diff. Eqn. 1, No. 4 (1989), 347–368.
I. Melbourne, P. Chossat & M. Golubitsky, Heteroclinic cycles involving periodic solutions in mode interactions with 0(2) symmetry, Proc. R. Soc. Edinburgh 113A (1989), 315–345.
D. Ruelle, Bifurcations in the presence of a symmetry group, Arch. Rational Mech. Anal. 51 (1973), 136–152.
F. Simonelli & J. Gollub, Surface wave mode interactions: effects of symmetry and degeneracy, J. Fluid Mech. 199 (1989), 471–494.
A. Vanderbauwhede, Local Bifurcation and Symmetry, Habilitation Thesis, Rijksuniver-siteit Ghent, 1980; Res. Notes in Math. 75, Pitman, Boston, 1982.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag New York Inc.
About this paper
Cite this paper
Golubitsky, M. (1991). Genericity, Bifurcation and Symmetry. In: Aris, R., Aronson, D.G., Swinney, H.L. (eds) Patterns and Dynamics in Reactive Media. The IMA Volumes in Mathematics and its Applications, vol 37. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3206-3_5
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3206-3_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7832-0
Online ISBN: 978-1-4612-3206-3
eBook Packages: Springer Book Archive