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New methods in the classification theory of bifurcation problems. (English) Zbl 0625.58016

Multiparameter bifurcation theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 56, 97-116 (1986).
[For the entire collection see Zbl 0587.00009.]
A bifurcation problem f(x,\(\lambda)\), \(f: {\mathbb{R}}^ n\times {\mathbb{R}}^ k\to {\mathbb{R}}^ p\), is called M-determined, M a fixed linear subspace of \(C^{\infty}({\mathbb{R}}^ n\times {\mathbb{R}}^ k,{\mathbb{R}}^ p)\), if f and g are contact equivalent whenever f-g\(\in M\). The author gives sufficient conditions for f to be M-determined. For intrinsic M (i.e., M invariant under the action of the group \({\mathcal B}\) whose orbits are the contact equivalence classes, the conditions are also necessary. The conditions come from J. W. Bruce, A. A. du Plessis, and C. T. C. Wall [Invent. Math. 88, 521-554 (1987; Zbl 0596.58005)], which deals with the analogous problem for singularities of maps. Examples illustrate how this result simplifies the work of classifying bifurcation problems. A second part of the paper develops some equivalence results that use directly the tangent space to an orbit of \({\mathcal B}\), rather than the reduced tangent space as in the work of M. Golubitsky and D. G. Schaeffer [Singularities and groups in bifurcation theory, Vol. I (1985; Zbl 0607.35004)]. The latter is easier to work with because it is an \({\mathcal E}_{x,\lambda}\) module.
Reviewer: St.Schecter

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems