On one scenario for changing the stability of invariant manifolds of singularly perturbed systems

O. S. Kipkaeva

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 30 (2024) no. 2, pp. 20-29

Résumé

The article is devoted to the peculiarities of stability change of slow invariant manifolds of singularly perturbed systems of ordinary differential equations. It should be noted that the change of stability of invariant manifolds can proceed according to different scenarios. In addition to two well-known scenarios of this phenomenon, one more scenario is considered in this paper. To demonstrate the peculiarities of the stability change of slow invariant manifolds under this scenario, a number of examples are proposed. The existence theorem of an exact invariant manifold with stability change for some class of singularly perturbed systems of ordinary differential equations is obtained.

Détail

Keywords: dynamical systems, invariant manifolds, stability, delayed stability loss, existence theorem.
Mots-clés : singular perturbations, canards, bifurcation

Bibliographie

[1] Strygin V.V., Sobolev V.A., Decomposition of motions by the integral manifolds method, Nauka, M., 256 pp. (In Russ.) | MR

[2] Shchepakina E., Sobolev V., Mortell M.P., Singular Perturbations. Introduction to system order reduction methods with applications, Springer, Cham-Berlin-Heidelber-London, 2014, 222 pp. | DOI | MR | Zbl

[3] Sobolev V.A., Shchepakina E.A., Reduction of models and critical phenomena in macrokinetics, FIZMATLIT, M., 2010, 320 pp. (In Russ.)

[4] Arnold V.I. et al., Bifurcation Theory, v. 5, VINITI, M., 1986, 218 pp. (In Russ.)

[5] Shchepakina E.A., “Two Forms of Stability Change for Integral Manifolds”, Differential Equations, 40:5 (2004), 766–769 (In English; original in Russian) | DOI | MR | Zbl

[6] Shchepakina E., Sobolev V., “Integral manifolds, canards and black swans”, Nonlinear Analysis: Theory, Methods Applications, 44 (2001), 897–908 | DOI | MR | Zbl

[7] Shchepakina E.A., “Stable/unstable slow integral manifolds in critical cases”, Journal of Physics: Conference Series, 811:1 (2017), 012016 | DOI | MR

[8] Benoit E., Callot J. L., Diener F., Diener M., “Chasse au canard”, Collectanea Mathematica, 31–32 (1981), 37–119 https://www.researchgate.net/publication/265548510_Chasse_au_canard | MR

[9] Gorelov G.N., Sobolev V.A. Mathematical modeling of critical phenomena in thermal explosion theory, Combustion and Flame, 87:2 (1991), 203–210 | DOI

[10] Gorelov G.N., Sobolev V.A., “Duck-trajectories in a thermal explosion problem”, Applied Mathematics Letters, 5:6 (1992), 3–6 | DOI | MR | Zbl

[11] Sobolev V.A., Shchepakina E.A., “Duck trajectories in a problem of combustion theory”, Differential Equations, 32:9 (1996), 1177–1186 | MR | Zbl

[12] Shchepakina E.A., “Black swans and canards in self-ignition problem”, Nonlinear Analysis: Real World Applications, 4:1 (2003), 45–50 | DOI | MR | Zbl

[13] Shchepakina E., Sobolev V., “Black swans and canards in laser and combustion models”, Singular perturbations and hysteresis, eds. M.P. Mortell, R.E. O'Malley, A. Pokrovskii, V.A. Sobolev, SIAM, Philadelphia, 2005, 207–255 | DOI | MR | Zbl

[14] Shchepakina E., “Canards and black swans in model of a 3-D autocatalator”, Journal of Physics: Conference Series, 22:1 (2005), 194–207 | DOI

[15] Shchepakina E., Korotkova O., “Condition for canard explosion in a semiconductor optical amplifier”, Journal of the Optical Society of America B: Optical Physics, 28:8 (2011), 1988–1993 | DOI | MR

[16] Shchepakina E., Korotkova O. Canard explosion in chemical and optical systems, Discrete and Continuous Dynamical Systems — Series B, 18:2 (2013), 495–512 | DOI | MR | Zbl

[17] Shchepakina E., Sobolev V., “Invariant surfaces of variable stability”, Journal of Physics: Conference Series, 727:1 (2016), 012016 pp. | DOI | MR | Zbl

[18] Neustadt A.I., “Asymptotic investigation of the loss of stability by an equilibrium as a pair of eigenvalues slowly cross the imaginary axis”, Uspekhi Matematicheskikh Nauk, 40:5 (1985), 190–191 (In English; original in Russian)

[19] Neustadt A.I., “Prolongation of the loss of stability loss in the case of dynamic bifurcations”, Differentsial'nye Uravneniya, 24:2, 226–233 | MR | MR

[20] Shishkova M.A. Examination of a system of differential equations with a small parameter in the highest derivatives, Doklady Akademii Nauk SSSR, 209:3 (1973), 576–579 (In Russ.) | MR | Zbl

[21] Shchetinina E.V., “A problem on the change of stability of integral manifolds”, Izvestija RAEN. Matematika. Matematicheskoe modelirovanie. Informatika i upravlenie, 3:3 (1999), 129–134 (In Russ.)

[22] Schneider K.R., Shchetinina E.V., Sobolev V.A., “Control of integral manifolds loosing their attractivity in time”, Journal of Mathematical Analysis and Applications, 315:2 (2006), 740–757 | DOI | MR | Zbl