Group symmetry of the Lyapunov–Schmidt branching equation and iterative methods in the problem of a bifurcation point

B. V. Loginov   ; N. A. Sidorov

Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 67-77

Résumé

Using group-theoretic methods (MR 80d: 58072, 83m: 58082), the authors construct the general form of the branching equation, symmetric with respect to fundamental representations of the rotation group, and on the basis of this form they propose an iterative method for calculating families of small branching solutions in a neighborhood of a bifurcation point, depending on free parameters.

Bibliographie

[1] Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR | Zbl

[2] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980 | MR | Zbl

[3] Yudovich V. I., “Svobodnaya konvektsiya i vetvlenie”, PMM, 31:1 (1967), 101–111 | Zbl

[4] Ter-Grigoryants G. K., “O vozniknovenii dvoyakoperiodicheskoi konvektsii v gorizontalnom sloe”, PMM, 1973, no. 1, 177–184

[5] Loginov B. V., Trenogin V. A., “Ob ispolzovanii gruppovykh svoistv dlya opredeleniya mnogoparametricheskikh semeistv reshenii nelineinykh uravnenii”, Matem. sb., 85 (127) (1971), 440–454 | MR | Zbl

[6] Loginov B. V., Trenogin V. A., “Ob ispolzovanii gruppovoi invariantnosti v teorii vetvleniya”, Differents. uravneniya, 11 (1975), 1709–1712 | MR | Zbl

[7] Loginov B. V., Teoriya vetvleniya reshenii nelineinykh uravnenii v usloviyakh gruppovoi invariantnosti, Izd-vo Fan, Tashkent, 1985 | MR

[8] Vanderbauwhede A., Local Bifurcation and Symmetry, Pitman, Boston, 1982 | MR | Zbl

[9] Loginov B. V., “On the construction of the general form of branching equation by its symmetry group”, Equadiff 7, Enlarged abstracts, Praha, 1989, 48–50

[10] Loginov B. V., Rakhmatova Kh. R., Yuldashev N., “O postroenii uravneniya razvetvleniya po ego gruppe simmetrii (kristallograficheskie gruppy)”, Uravneniya smeshannogo tipa i zadachi so svobodnoi granitsei, Izd-vo Fan, Tashkent, 1987, 183–195 | MR

[11] Trenogin V. A., Sidorov N. A., Loginov B. V., “Potentiality, group symmetry and bifurcation in the theory of branching equation”, Differential and Integral Equations. Ohio, USA, 3:1 (1990), 145–154 | MR | Zbl

[12] Loginov B. V., Sabirova S. G., “O vetvlenii periodicheskikh reshenii nelineino vozmuschennykh ellipticheskikh uravnenii”, Izv. AN UzSSR, 1988, no. 4, 32–37 | MR | Zbl

[13] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[14] Ovsyannikov L. V., Lektsii po teorii gruppovykh svoistv differentsialnykh uravnenii, Izd-vo NGU, Novosibirsk, 1966

[15] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983 | MR

[16] Sidorov N. A., Obschie voprosy regulyatsii v zadachakh teorii vetvleniya, Izd-vo Irkut. un-ta, Irkutsk, 1982 | MR | Zbl

[17] Sidorov N. A., “O postroenii razvetvlyayuschikhsya reshenii, zavisyaschikh ot svobodnykh parametrov, metodom posledovatelnykh priblizhenii”, Mezhdunarodnaya shkola-seminar po metodam optimizatsii i ikh prilozheniyam (Irkutsk, 1989), SO AN SSSR, SEI, 180–181

[18] Trenogin V. A., Sidorov N. A., Loginov B. V., “Uravnenie razvetvleniya: potentsialnost, bifurkatsiya, simmetriya”, DAN SSSR, 309:2 (1989), 286–289 | MR

[19] Bryuno A. D., Lokalnyi metod nelineinogo analiza differentsialnykh uravnenii, Nauka, M., 1979 | MR

[20] Golubitsky M., Schaeffer D., Stewart I., Singularities and Groups in Bifurcation Theory, 1, Springer, N. Y., 1985; 2, 1988 | Zbl