Geodesic
Self-excited wave processes in chains of diffusion-linked delay equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 2, pp. 297-343 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A new mathematical object is introduced, a scalar non-linear difference-differential equation with time delay which is a certain modification of the Hutchinson equation well-known in ecology. It is shown that the buffering phenomenon occurs in a one-dimensional chain of diffusion-linked equations of this type. Namely, as the number of links grows in a way compatible with a decrease of the diffusion coefficient, the number of co-existing stable periodic solutions of the system increases without limit. Bibliography: 15 titles.
Keywords: modified Hutchinson equation, self-excited wave processes, relaxation cycle, asymptotic behaviour, stability. 
@article{RM_2012_67_2_a3,
     author = {A. Yu. Kolesov and N. Kh. Rozov},
     title = {Self-excited wave processes in chains of diffusion-linked delay equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {297--343},
     year = {2012},
     volume = {67},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2012_67_2_a3/}
}
TY  - JOUR
AU  - A. Yu. Kolesov
AU  - N. Kh. Rozov
TI  - Self-excited wave processes in chains of diffusion-linked delay equations
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2012
SP  - 297
EP  - 343
VL  - 67
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/RM_2012_67_2_a3/
LA  - en
ID  - RM_2012_67_2_a3
ER  - 
%0 Journal Article
%A A. Yu. Kolesov
%A N. Kh. Rozov
%T Self-excited wave processes in chains of diffusion-linked delay equations
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2012
%P 297-343
%V 67
%N 2
%U http://geodesic.mathdoc.fr/item/RM_2012_67_2_a3/
%G en
%F RM_2012_67_2_a3
A. Yu. Kolesov; N. Kh. Rozov. Self-excited wave processes in chains of diffusion-linked delay equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 2, pp. 297-343. http://geodesic.mathdoc.fr/item/RM_2012_67_2_a3/

[1] E. F. Mischenko, L. S. Pontryagin, “Periodicheskie resheniya sistem differentsialnykh uravnenii, blizkie k razryvnym”, Dokl. AN SSSR, 102:5 (1955), 889–891 | MR | Zbl

[2] E. F. Mischenko, “Asimptoticheskoe vychislenie periodicheskikh reshenii sistem differentsialnykh uravnenii, soderzhaschikh malye parametry pri proizvodnykh”, Izv. AN SSSR. Ser. matem., 21:5 (1957), 627–654 | MR | Zbl

[3] E. F. Mischenko, L. S. Pontryagin, “Dokazatelstvo nekotorykh asimptoticheskikh formul dlya reshenii differentsialnykh uravnenii s malym parametrom”, Dokl. AN SSSR, 120:5 (1958), 967–969 | MR | Zbl

[4] E. F. Mischenko, L. S. Pontryagin, “Vyvod nekotorykh asimptoticheskikh otsenok dlya reshenii differentsialnykh uravnenii s malym parametrom pri proizvodnykh”, Izv. AN SSSR. Ser. matem., 23:5 (1959), 643–660 | MR | Zbl

[5] L. S. Pontryagin, “Asimptoticheskoe povedenie reshenii sistem differentsialnykh uravnenii s malym parametrom pri vysshikh proizvodnykh”, Izv. AN SSSR. Ser. matem., 21:5 (1957), 605–626 | MR | Zbl

[6] E. F. Mishchenko, N. Kh. Rozov, Differential equations with small parameters and relaxation oscillations, Math. Concepts Methods Sci. Engrg., 13, Plenum Press, New York, London, 1980, x+228 pp. | MR | MR | Zbl | Zbl

[7] E. F. Mischenko, Yu. C. Kolesov, A. Yu. Kolesov, N. Kh. Rozov, Periodicheskie dvizheniya i bifurkatsionnye protsessy v singulyarno vozmuschennykh sistemakh, Nauka, Fizmatlit, M., 1995, 336 pp. | MR | Zbl

[8] A. Yu. Kolesov, Yu. S. Kolesov, “Relaxation oscillations in mathematical models of ecology”, Proc. Steklov Inst. Math., 199 (1995), 1–126 | MR | Zbl | Zbl

[9] A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “A relay with delay and its $C^1$-approximation”, Proc. Steklov Inst. Math., 216 (1997), 119–146 | MR | Zbl

[10] G. E. Hutchinson, “Circular causal systems in ecology”, Ann. N. Y. Acad. Sci., 50 (1948), 221–246 | DOI

[11] E. M. Wright, “A non-linear difference-differential equation”, J. Reine Angew. Math., 194 (1955), 66–87 | DOI | MR | Zbl

[12] G. S. Jones, “Asymptotic behavior and periodic solutions of a nonlinear differential-difference equation”, Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 879–882 | DOI | MR | Zbl

[13] J. Hale, Theory of functional differential equations, Appl. Math. Sci., 3, 2nd ed., New York–Heidelberg, Springer-Verlag, 1977, x+365 pp. | MR | MR | Zbl | Zbl

[14] A. Yu. Kolesov, N. Kh. Rozov, Invariantnye tory nelineinykh volnovykh uravnenii, Fizmatlit, M., 2004, 408 pp.

[15] E. F. Mischenko, V. A. Sadovnichii, A. Yu. Kolesov, N. Kh. Rozov, Avtovolnovye protsessy v nelineinykh sredakh s diffuziei, Fizmatlit, M., 2005, 432 pp.