Bifurcations of critical tori for functionals with 3-circular symmetry. (English. Russian original) Zbl 0985.37047

Gnezdilov, A. V.

Funct. Anal. Appl. 34, No. 1, 67-69 (2000); translation from Funkts. Anal. Prilozh. 34, No. 1, 83-86 (2000).

Bifurcations of sets of critical points of a smooth functional defined on a Banach manifold \(M\) and invariant with respect to a smooth action of an \(m\)-dimensional torus have a wide range of applications. This paper deals with this problem in the case \(m=3\). In particular, the author gives a complete description of the bifurcation of Morse critical orbits from a point of minimum with a 3-circular pleat singularity.

Reviewer: Messoud Efendiev (Berlin)

MSC:

37G40	Dynamical aspects of symmetries, equivariant bifurcation theory
58K45	Singularities of vector fields, topological aspects
37G10	Bifurcations of singular points in dynamical systems

Keywords:

tori; bifurcation; critical point; Morse critical orbit; group action

References:

[1]	B. A. Strukov and A. P. Levanyuk, Physical Foundations of Ferroelectric Phenomena in Crystals [in Russian], Nauka, Moscow 1995. · Zbl 0899.00012
[2]	B. M. Darinskii and Yu. I. Sapronov, Izv. Vyssh. Uchebn. Zaved., Ser. Mat.,41, No. 2, 35–46 (1997)
[3]	Yu. I. Sapronov, Usp. Mat. Nauk,51, No. 1, 101–132 (1996).
[4]	D. Siersma, Quart. J. Math. Oxford Ser. (2),32, No. 125, 119–127 (1981). · Zbl 0449.58008 · doi:10.1093/qmath/32.1.119
[5]	V. I. Arnold, A. N. Varchenko, and S. M. Gussein-Zade, Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics, and Wave Fronts, Birkhauser, Boston, 1985.
[6]	Yu. I. Sapronov, Mat. Sb.,180, No. 10, 1299–1310 (1989).
[7]	A. V. Gnezdilov, Pontryagin Readings-VIII, Abstracts Voronezh, Voronezh State University, 1997, p. 36.
[8]	A. V. Gnezdilov, Trudy Matem. Fakul’teta VGU (Novaya Seriya),2, (18), 19–26 (1997).