Geodesic
On immediate-delayed exchange of stabilities and periodic forced canards
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 1, pp. 46-61 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Singularly perturbed nonautonomous ordinary differential equations are studied for which the associated equations have equilibrium states consisting of at least two intersecting curves, which leads to exchange of stabilities of these equilibria. The asymptotic method of differential equations is used to derive conditions under which initial value problems have solutions characterized by immediate and delayed exchange of stabilities. These results are then used to prove the existence of periodic canard solutions. 
@article{ZVMMF_2008_48_1_a3,
     author = {N. N. Nefedov and K. R. Schneider},
     title = {On immediate-delayed exchange of stabilities and periodic forced canards},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {46--61},
     year = {2008},
     volume = {48},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_1_a3/}
}
TY  - JOUR
AU  - N. N. Nefedov
AU  - K. R. Schneider
TI  - On immediate-delayed exchange of stabilities and periodic forced canards
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 46
EP  - 61
VL  - 48
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_1_a3/
LA  - en
ID  - ZVMMF_2008_48_1_a3
ER  - 
%0 Journal Article
%A N. N. Nefedov
%A K. R. Schneider
%T On immediate-delayed exchange of stabilities and periodic forced canards
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 46-61
%V 48
%N 1
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_1_a3/
%G en
%F ZVMMF_2008_48_1_a3
N. N. Nefedov; K. R. Schneider. On immediate-delayed exchange of stabilities and periodic forced canards. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 1, pp. 46-61. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_1_a3/

[1] Marsden J. E., “Qualitative methods in bifurcation theory”, Bull. Amer. Math. Soc., 84 (1978), 1125–1148 | DOI | MR | Zbl

[2] Ruelle D., Elements of differentiable dynamics and bifurcation theory, Bosten Acad. Press, Inc., 1989 | MR | Zbl

[3] Shilnikov L. P., Shilnikov A. L., Turaev D. V., Chua L. O., Methods of qualitative theory in nonlinear dynamics, Part II, World Scient., Singapore, 2001 | MR | Zbl

[4] Benoît E. (ed.), Dynamic bifurcation, Lect. Not. Math., 1493, Springer, New York, 1991

[5] Lebovitz N. R., Schaar R. J., “Exchance of stabilities in autonomous systems”, Stud. Appl. Math., 54 (1975), 229–260 | MR | Zbl

[6] Arnold V. I., Afraimovich V. S., Il'yashenko Yu. S., Shil'nikov L. P., Theory of bifurcations dynamical systems, Encyclopedia of Math. Sci., 5, Springer, New York, 1994

[7] Butuzov V. F., Nefedov H. H., “Singulyarno vozmuschennaya kraevaya zadacha dlya uravneniya vtorogo poryadka v sluchae smeny ustoichivosti”, Matem. zametki, 63:3 (1998), 354–362 | MR | Zbl

[8] Butuzov V. F., Nefedov N. N., Schneider K. R., “Singularly perturbed boundary value problems in case of exchance of stabilities”, J. Math. Analys. and Appl., 229 (1999), 543–562 | DOI | MR | Zbl

[9] Butuzov V. F., Nefedov H. H., Shnaider K. P., “Differentsialnye uravneniya. Singulyarnye vozmuscheniya”, Itogi nauki i tekhn. Ser. sovrem. matem. i ee prilozh. Tematich. obzory, 109, VINITI, M., 2003, 1–144

[10] Butiizov V. F., Nefedov N. N., Schneider K. R., “Singularly perturbed reaction-diffusion systems in cases of exchange of stabilities”, Nat. Res. Model., 13:2 (2000), 247–269 | MR

[11] Dumortier F., Smits B., “Transition time analysis in singularly perturbed boundary value problems”, Trans. Amer. Math. Soc., 347 (1995), 4129–4145 | DOI | MR | Zbl

[12] Dumortier F., Roussarie R., Canard cycles and center manifolds, Mem. Amer. Math. Soc., 577, 1996 | MR | Zbl

[13] Gorelov G. N., Soholev V. A., “Duck-trajectories in a thermal explosion problem”, Appl. Math. Letts., 5:6 (1992), 3–6 | DOI | MR | Zbl

[14] Gorelov G. N., Soholev V. A., “Mathematical modeling of critical phenomena in thermal explosion theory”, Combust. Flame, 87 (1991), 203–210 | DOI

[15] Kolesov A. Yu., Rozov N. Kh., ““Okhota na utok” v issledovanii singulyarno vozmuschennykh kraevykh zadach”, Differents. ur-niya, 35:7 (1995), 1356–1365 | MR

[16] Kolesov A. Yu., Rozov N. Kh., “Problema “Buridanova osla” v relaksatsionnykh sistemakh s odnoi medlennoi peremennoi”, Matem. zametki, 65:1 (1999), 153–156 | MR | Zbl

[17] Krupa M., Szmolyan P., “Extending geometric singular perturbation theory to non-hyperbolic points-fold and canard points in two dimensions”, SIAM J. Math. Analys., 33 (2001), 286–314 | DOI | MR | Zbl

[18] Nefedov N. N., Schneider K. R., Delayed exchange of stabilities in singularly perturbed systems, Preprint No. 270, Weierstraß-Inst. fur Angewandte Analysis und Stochastik, Berlin, 1996

[19] Nefedov N. N., Schneider K. R., “Immediate exchange of stabilities in singularly perturbed systems”, Different. Integr. Equat., 12 (1999), 583–599 | MR | Zbl

[20] Neishtadt A. M., “O zapazdyvanii poteri ustoichivosti pri dinamicheskikh bifurkatsiyakh. I”, Differents. ur-niya, 23 (1987), 2060–2067 | MR

[21] Neishtadt A. I., “O zapazdyvanii poteri ustoichivosti pri dinamicheskikh bifurkatsiyakh. II”, Differents. ur-niya, 24 (1988), 226–233 | MR

[22] Shishkova M. A., “Issledovanie odnoi sistemy differentsialnykh uravnenii s malym parametrom pri starshikh proizvodnykh”, Dokl. AN SSSR, 209 (1973), 576–579 | Zbl

[23] Shchepakina E. A., Soholev V. A., “Integral manifolds, canards and black swans”, Nonlinear. Analys. Theory, Methods, Applications, 44 (2001), 897–908 | DOI | MR | Zbl

[24] Chaplygin C. A., Novyi metod priblizhennogo integrirovaniya differentsialnykh uravnenii, Gostekhteorizdat, M.-L., 1950

[25] Pao C. V., Nonlinear parabolic and equations, Plenum Press, New York, London, 1992 | MR

[26] Vasil'eva A. B., Butuzov V. F., Kalachev L. V., The boundary function method for singular perturbation problems, SIAM Studies Appl. Math., Philadelphia, 1995 | MR

[27] Tikhonov A. N., “Sistemy differentsialnykh uravnenii, soderzhaschie malye parametry”, Matem. sb., 31(73):3 (1952), 575–586 | Zbl

[28] Nefedov H. H., “Metod differentsialnykh neravenstv dlya nekotorykh klassov nelineinykh singulyarno vozmuschennykh zadach s vnutrennimi sloyami”, Differents. ur-niya, 31:7 (1995), 1132–1139