Geodesic
Study of main scenarios of bifurcation for functional differential time-delay equations
Ufa mathematical journal, Tome 6 (2014) no. 2, pp. 102-110 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper, we study the main bifurcation scenarios for functional differential time delay equations with periodic right side and for nonlinear autonomous equations with aftereffect. The main tool is the operator method for studying multi-parameter bifurcation leading us to new sufficient bifurcation conditions and allowing us to obtain the approximate formulae for appearing solutions. As applications, we consider the problems on bifurcation points for the modifications of Duffing equation and Hutchinson–Wright equation.
Keywords: functional differential equations, time-delay system, dynamical systems, operator method, functionalization parameter, asymptotic formula.
Mots-clés : bifurcation 
@article{UFA_2014_6_2_a7,
     author = {M. G. Yumagulov and D. A. Yakshibaeva},
     title = {Study of main scenarios of bifurcation for functional differential time-delay equations},
     journal = {Ufa mathematical journal},
     pages = {102--110},
     year = {2014},
     volume = {6},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2014_6_2_a7/}
}
TY  - JOUR
AU  - M. G. Yumagulov
AU  - D. A. Yakshibaeva
TI  - Study of main scenarios of bifurcation for functional differential time-delay equations
JO  - Ufa mathematical journal
PY  - 2014
SP  - 102
EP  - 110
VL  - 6
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UFA_2014_6_2_a7/
LA  - en
ID  - UFA_2014_6_2_a7
ER  - 
%0 Journal Article
%A M. G. Yumagulov
%A D. A. Yakshibaeva
%T Study of main scenarios of bifurcation for functional differential time-delay equations
%J Ufa mathematical journal
%D 2014
%P 102-110
%V 6
%N 2
%U http://geodesic.mathdoc.fr/item/UFA_2014_6_2_a7/
%G en
%F UFA_2014_6_2_a7
M. G. Yumagulov; D. A. Yakshibaeva. Study of main scenarios of bifurcation for functional differential time-delay equations. Ufa mathematical journal, Tome 6 (2014) no. 2, pp. 102-110. http://geodesic.mathdoc.fr/item/UFA_2014_6_2_a7/

[1] Bellman R., Kuk K. L., Differentsialno-raznostnye uravneniya, Mir, M., 1967, 548 pp. | MR | Zbl

[2] Myshkis A. D., Lineinye differentsialnye uravneniya s zapazdyvayuschim argumentom, Nauka, M., 1972, 352 pp. | MR | Zbl

[3] Kheil Dzh., Teoriya funktsionalno-differentsialnykh uravnenii, Mir, M., 1984, 421 pp. | MR

[4] B. Balachandran, T. Kalmar-Nagy, D. Gilsinn, Delay Differential Equations. Recent Advances and New Directions, Springer, New York, NY, 2009 | MR | Zbl

[5] Kolesov Yu. S., “Obosnovanie metoda kvazinormalnykh form dlya uravneniya Khatchinsona s malym koeffitsientom diffuzii”, Izvestiya RAN. Ser. matem., 65:4 (2001), 111–132 | DOI | MR | Zbl

[6] D. Roose, R. Szalai, “Continuation and bifurcation analysis of delay differential equations”, Numerical continuation methods for dynamical systems, Springer, Dordrecht, 2007, 359–399 | DOI | MR | Zbl

[7] D. Schley, “Bifurcation and stability of periodic solutions of differential equations with state-dependent delays”, European Journal of Applied Mathematics, 14:1 (2003), 3–14 | DOI | MR | Zbl

[8] Kamenskii M. I., Lysakova Yu. V., Nistri P., “O bifurkatsii periodicheskikh reshenii dlya funktsionalno-differentsialnykh uravnenii neitralnogo tipa s malym zapazdyvaniem”, Avtomatika i telemekhanika, 2008, no. 12, 41–46 | MR | Zbl

[9] Vyshinskii A. A., Ibragimova L. S, Murtazina S. A., Yumagulov M. G, “Operatornyi metod priblizhennogo issledovaniya pravilnoi bifurkatsii v mnooparametricheskikh dinamicheskikh sistemakh”, Ufimskii matematicheskii zhurnal, 2:4 (2010), 3–26 | Zbl

[10] Krasnoselskii M. A., Operator sdviga po traektoryaim differentsialnykh uravnenii, Nauka, M., 1966, 332 pp. | MR | Zbl

[11] Khessard B., Kazarinov N., Ven I., Teoriya i prilozheniya bifurkatsii rozhdeniya tsikla, Mir, M., 1985, 280 pp. | MR

[12] Ibragimova L. S., Yumagulov M. G., “Funktsionalizatsiya parametra i ee prilozheniya v zadache o lokalnykh bifurkatsiyakh dinamicheskikh sistem”, Avtomatika i telemekhanika, 2007, no. 4, 3–12 | MR | Zbl

[13] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975, 512 pp. | MR

[14] Krasnoselskii M. A., Vainikko G. M., Zabreiko P. P., Rutitskii Ya. B., Stetsenko V. Ya., Priblizhennoe reshenie operatornykh uravnenii, Nauka, M., 1969, 456 pp. | MR