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A discussion of symmetry and symmetry breaking. (English) Zbl 0535.58010

Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 499-515 (1983).
[For the entire collection see Zbl 0509.00008.] In this paper the authors review some theoretical aspects of their singularity theory approach to imperfect steady state bifurcation theory and in particular to symmetric problems [see Commun. Math. Phys. 67, 205-232 (1979; Zbl 0467.58019)]. First they review the basic theory for one state variable, introducing such concepts as equivalence between bifurcation problems and universal unfoldings. Next they indicate how these concepts are to be modified when symmetry comes into play; they also show that in certain cases it is possible to incorporate stability considerations in the theory. The paper concludes with a discussion of spontaneous symmetry breaking and symmetry breaking in the equations; the conclusion is that although a number of particular cases have been studied, there still remains to build up a general theory covering these aspects of bifurcation theory.
Reviewer: A.Vanderbauwhede

MSC:

58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
37G99 Local and nonlocal bifurcation theory for dynamical systems