Geodesic
Discrete autowaves in systems of delay differential--difference equations in ecology
Informatics and Automation, Mathematical control theory and differential equations, Tome 277 (2012), pp. 101-143.

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We propose a theory of relaxation oscillations for a nonlinear scalar delay differential-difference equation that represents a modification of the well-known Hutchinson equation in ecology. In particular, we establish that a one-dimensional chain of diffusively coupled equations of this type exhibits the well-known buffer phenomenon. Namely, under an increase in the number of links in the chain and a consistent decrease in the coupling constant, the number of coexisting stable periodic motions indefinitely increases. 
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A. Yu. Kolesov; N. Kh. Rozov. Discrete autowaves in systems of delay differential--difference equations in ecology. Informatics and Automation, Mathematical control theory and differential equations, Tome 277 (2012), pp. 101-143. http://geodesic.mathdoc.fr/item/TRSPY_2012_277_a7/

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

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

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

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

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

[6] Mischenko E.F., Rozov N.Kh., Differentsialnye uravneniya s malym parametrom i relaksatsionnye kolebaniya, Nauka, M., 1975 | MR

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

[8] Kolesov A.Yu., Kolesov Yu.S., Relaksatsionnye kolebaniya v matematicheskikh modelyakh ekologii, Tr. MIAN, 199, Nauka, M., 1993 | MR | Zbl

[9] Kolesov A.Yu., Mischenko E.F., Rozov N.Kh., “Rele s zapazdyvaniem i ego $C^1$-approksimatsiya”, Tr. MIAN, 216, 1997, 126–153 | MR | Zbl

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

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

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

[13] Hale J., Theory of functional differential equations, Springer, New York, 1977 | MR | Zbl

[14] Kolesov A.Yu., Mischenko E.F., Rozov N.Kh., “Ob odnoi modifikatsii uravneniya Khatchinsona”, ZhVMiMF, 50:12 (2010), 2099–2112 | MR | Zbl

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

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