Group analysis methods for construction and investigation of the bifurcation equation. (English) Zbl 0795.58035

The author presents the methods of group analysis of differential equations for the construction of the general form of the bifurcation equation.


37G99 Local and nonlocal bifurcation theory for dynamical systems
37C80 Symmetries, equivariant dynamical systems (MSC2010)
20H15 Other geometric groups, including crystallographic groups
Full Text: EuDML


[1] B. V. Loginov: Branching theory of solutions of nonlinear equations in conditions of group invariance. FAN, Tashkent, 1985. · Zbl 0593.58028
[2] B. V. Loginov V. A. Trenogin: On the use of group invariance in bifurcation theory. Differential Equations 11 no. 8 (1975), 1518-1521. · Zbl 0314.47039
[3] J. Dieudonne J. Carrel D. Mamfold: Geometric theory of invariants. Mir, Moskva, 1974.
[4] L. V. Ovsjannikov: Group analysis of differential equations. Nauka, Moskva, 1978)
[5] L. V. Ovsjanikov: Lectures on the theory of group properties of differential equations. NGU, Novosibirsk, 1966.
[6] N. Ch. Ibragimov: Groups of transformations in mathematical physics. Nauka, Moskva, 1983.
[7] B. V. Loginov, Ch. R. Rachmatova N. Juldashev: On construction of bifurcation equation by its symmetry group. (cristallographic groups), Mixed type equations and free boundary value problems, FAN, Tashkent, 1985, pp. 183-195.
[8] B. V. Loginov A. O. Kusnetsov: On construction of periodic solutions of three-dimensional problem of capillary-gravitational waves over a flat bottom in the case of high degeneracy. Proc. ICNO-11, Budapest, Aug. 17-23, 1987, Budapest, 1987, pp. 669-671.
[9] B. V. Loginov S. G. Sabirova: On the bifurcation of periodic solutions of nonlinearly perturbed elliptic equations. Izvestija AN Uzbek SSR, fiz.-mat, nauki no. 4 (1988), 32-37. · Zbl 0703.35016
[10] M. M. Vainberg V. A. Trenogin: Branching theory of solutions of nonlinear equations. Nauka, Moskva, 1969 (In Russian.); in · Zbl 0274.47033
[11] Ch. Kittel: Introduction into solid states physics. Nauka, Moskva, 1978.
[12] W. Heine: Group theory in quantum mechanics. IL, Moskva, 1969.
[13] G. Ja. Ljubarsky: Group theory and its applications in physics. GITTL, Moskva, 1958.
[14] Ph. Delanō: Bifurcation for Monge-Ampere equations on flat tori. Manuscripta mathematica no. 1 (1983), 29-45. · Zbl 0536.58034
[15] T. Aubin: Metriques Riemanniennes et Courbure. J. Diff. Geometry 48(1970), 383-424. · Zbl 0212.54102
[16] I. M. Vinogradov: Foundations of the number theory. Nauka, Moskva, 1981. · Zbl 0547.10001
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