Geodesic
Similarity and difference in the dynamics of plane and 3-dimensional non-linear waves
Sbornik. Mathematics, Tome 196 (2005) no. 2, pp. 201-229 
Yu. S. Kolesov; A. E. Khar'kov. Similarity and difference in the dynamics of plane and 3-dimensional non-linear waves. Sbornik. Mathematics, Tome 196 (2005) no. 2, pp. 201-229. http://geodesic.mathdoc.fr/item/SM_2005_196_2_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The method of quasinormal forms is used to demonstrate the mild loss of stability by travelling waves on a circle and a torus. The dynamics in the case of a small elastic-constraint coefficient is studied in a model example. The results obtained and an explicit difference scheme developed expressly for the problem are used in a series of numerical experiments. As a result one distinguishes a subdomain of the parameter space in which the most important features of the dynamics on a torus and a circle are related to self-organisation regimes, where self-organisation regimes on a circle are versions of certain similar regimes on a torus.

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

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

[4] Kolesov Yu. S., “Asimptotika i ustoichivost nelineinykh parametricheskikh kolebanii singulyarno vozmuschennogo telegrafnogo uravneniya”, Matem. sb., 186:10 (1995), 57–72 | MR | Zbl

[5] Samoilenko A. M., Elementy matematicheskoi teorii mnogochastotnykh kolebanii. Invariantnye tory, Nauka, M., 1987 | MR

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[7] Malomed B. A., Scheuer J., “Stable and chaotic solutions of the complex Ginzburg–Landau equation with periodic boundary conditions”, Physica D, 161 (2002), 102–115 | DOI | MR | Zbl

[8] Kolesov Yu. S., Maiorov V. V., “Prostranstvennaya i vremennaya samoorganizatsiya v odnovidovom biotsenoze”, Dinamika biologicheskikh populyatsii, ed. V. D. Goryachenko, Izd-vo Gorkovskogo un-ta, Gorkii, 1986, 3–18

[9] Kolesov Yu. S., “Paradoksalnye svoistva attraktorov nelineinykh volnovykh uravnenii v ploskikh oblastyakh”, Dokl. RAN, 187:3 (2002), 443–445 | MR

[10] Kolesov Yu. S., “Osobennosti bifurkatsionnoi problemy dlya volnovogo uravneniya v uzkoi scheli”, Dokl. RAN, 367:4 (1999), 442–444 | MR | Zbl