Geodesic
Impact of quadratic non-linearity on the dynamics
Sbornik. Mathematics, Tome 193 (2002) no. 1, pp. 93-118
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For the non-linear telegraph equation with homogeneous Dirichlet or Neumann conditions at the end-points of a finite interval the question of the existence and the stability of time-periodic solutions bifurcating from the zero equilibrium state is considered. The dynamics of these solutions under a change of the diffusion coefficient (that is, the coefficient of the second derivative with respect to the space variable) is investigated. For the Dirichlet boundary conditions it is shown that this dynamics substantially depends on the presence – or the absence – of quadratic terms in the non-linearity. More precisely, it is shown that a quadratic non-linearity results in the occurrence, under an unbounded decrease of diffusion, of an infinite sequence of bifurcations of each periodic solution. En route, the related issue of the limits of applicability of Yu.S. Kolesov's method of quasinormal forms to the construction of self-oscillations in singularly perturbed hyperbolic boundary value problems is studied. 
@article{SM_2002_193_1_a3,
     author = {A. Yu. Kolesov and N. Kh. Rozov},
     title = {Impact of quadratic non-linearity on the~dynamics},
     journal = {Sbornik. Mathematics},
     pages = {93--118},
     year = {2002},
     volume = {193},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2002_193_1_a3/}
}
TY  - JOUR
AU  - A. Yu. Kolesov
AU  - N. Kh. Rozov
TI  - Impact of quadratic non-linearity on the dynamics
JO  - Sbornik. Mathematics
PY  - 2002
SP  - 93
EP  - 118
VL  - 193
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_2002_193_1_a3/
LA  - en
ID  - SM_2002_193_1_a3
ER  - 
%0 Journal Article
%A A. Yu. Kolesov
%A N. Kh. Rozov
%T Impact of quadratic non-linearity on the dynamics
%J Sbornik. Mathematics
%D 2002
%P 93-118
%V 193
%N 1
%U http://geodesic.mathdoc.fr/item/SM_2002_193_1_a3/
%G en
%F SM_2002_193_1_a3
A. Yu. Kolesov; N. Kh. Rozov. Impact of quadratic non-linearity on the dynamics. Sbornik. Mathematics, Tome 193 (2002) no. 1, pp. 93-118. http://geodesic.mathdoc.fr/item/SM_2002_193_1_a3/

[1] Vitt A. A., “Raspredelennye avtokolebatelnye sistemy”, Zhurn. tekhnich. fiziki, 4:1 (1934), 144–157

[2] Migulin V. V., Medvedev V. I., Mustel E. R., Parygin V. N., Osnovy teorii kolebanii, Nauka, M., 1988

[3] Landa P. S., Nelineinye kolebaniya i volny, Nauka, M., 1997 | MR

[4] Kolesov A. Yu., Rozov N. Kh., “Asimptoticheskaya teoriya kolebanii v sisteme Vitta”, Sovr. matem. i ee prilozh., 67, VINITI, M., 1999, 5–68 | MR

[5] Kolesov Yu. S., “Zadacha parazit-khozyain”, Dinamika biologicheskikh populyatsii, Mezhvuz. sb., Gorkovskii un-t, Gorkii, 1984, 28–34 | MR

[6] Kolesov Yu. S., “Ob odnoi bifurkatsionnoi teoreme v teorii avtokolebanii raspredelennykh sistem”, Differents. uravneniya, 21:10 (1985), 1709–1713 | MR | Zbl

[7] Kolesov Yu. S., “Metod kvazinormalnykh form v zadache ob ustanovivshikhsya rezhimakh parabolicheskikh sistem s maloi diffuziei”, Ukr. matem. zhurn., 39:1 (1987), 27–34 | MR

[8] Kolesov A. Yu., Kolesov Yu. S., “Bifurkatsiya avtokolebanii singulyarno vozmuschennogo volnovogo uravneniya”, Dokl. AN SSSR, 315:2 (1990), 281–283 | MR | Zbl

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

[10] Kolesov A. Yu., Mischenko E. F., Rozov N. Kh., Asimptoticheskie metody issledovaniya periodicheskikh reshenii nelineinykh giperbolicheskikh uravnenii, Tr. MIAN, 222, Nauka, M., 1998

[11] Bogolyubov N. N., Mitropolskii Yu. A., Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR

[12] Vasileva A. B., Kaschenko S. A., Kolesov Yu. S., Rozov N. Kh., “Bifurkatsiya avtokolebanii nelineinykh parabolicheskikh uravnenii s maloi diffuziei”, Matem. sb., 130:4 (1986), 488–499 | MR | Zbl

[13] Kolesov Yu. S., Maiorov V. V., “Novyi metod issledovaniya ustoichivosti reshenii lineinykh differentsialnykh uravnenii s blizkimi k postoyannym pochti periodicheskimi koeffitsientami”, Differents. uravneniya, 10:10 (1974), 1778–1788 | MR | Zbl

[14] Kolesov A. Yu., “Ustoichivost avtokolebanii telegrafnogo uravneniya, bifurtsiruyuschikh iz sostoyaniya ravnovesiya”, Matem. zametki, 51:2 (1992), 59–65 | MR | Zbl

[15] Kolesov A. Yu., “Parametricheskie kolebaniya reshenii telegrafnogo uravneniya s umerenno maloi diffuziei”, Sib. matem. zhurn., 33:6 (1992), 79–86 | MR | Zbl

[16] Kambulov V. F., Kolesov A. Yu., Rozov N. Kh., “Suschestvovanie i ustoichivost bystro ostsilliruyuschikh tsiklov u nelineinogo telegrafnogo uravneniya”, ZhVM i MF, 38:8 (1998), 1287–1300 | MR | Zbl

[17] Kolesov Yu. S., “Bifurkatsiya invariantnykh torov parabolicheskikh sistem s maloi diffuziei”, Matem. sb., 184:3 (1993), 121–136 | Zbl

[18] Kambulov V. F., Kolesov A. Yu., Rozov N. Kh., “Bifurkatsiya prostranstvenno neodnorodnykh tsiklov u nelineinogo volnovogo uravneniya s maloi diffuziei”, Tr. MMO, 59, URSS, M., 1998, 197–220 | MR | Zbl

[19] Vasileva A. B., Butuzov V. F., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973 | MR | Zbl