Qualitative analysis of one singularly perturbed system of differential equations with a small parameter
https://doi.org/10.25587/2411-9326-2024-3-15-27
Abstract
We consider a singularly perturbed system of ordinary differential equations with a small parameter, in which different-scale variables are involved. The necessary information about the method of integral manifolds is presented, such as slow surface (ε = 0), sheets of slow surfaces, integral manifold (ε ̸= 0), its sheets, and asymptotic expansion of a slow integral manifold in powers of ε. As an example, a qualitative analysis of one singularly perturbed system with a small parameter is carried out in the work.
About the Authors
L. I. Kononenko
Sobolev Institute of Mathematics
Russian Federation
Russian Federation
Larisa I. Kononenko
4 Koptyug Avenue, 630090 Novosibirsk
E. P. Volokitin
Sobolev Institute of Mathematics
Russian Federation
Russian Federation
Evgenii P. Volokitin
4 Koptyug Avenue, 630090 Novosibirsk
References
1. Mitropolsky Yu. A. and Lykova O. B., Integral Manifolds in Nonlinear Mechanics [in Russian], Nauka, Moscow (1963).
2. Vasil’eva A. V. and Butuzov V. F., Singularly Perturbed Equations in Critical Cases [in Russian], Nauka, Moscow (1978).
3. Goldstein V. M. and Sobolev V. A., Qualitative Analysis of Singularly Perturbed Systems [in Russian], Inst. Mat., Novosibirsk (1988).
4. Mishchenko E. F. and Rozov N. Kh., Differential Equations with Small Parameter and Relaxation Oscillations [in Russian], Nauka, Moscow (1975).
5. Strygin V. V. and Sobolev V. A., Separation of Motions Using the Method of Integral Manifolds [in Russian], Nauka, Moscow (1988).
6. Kononenko L. I., “On the smoothness of slow surfaces of singularly perturbed systems [in Russian],” Sib. Zh. Ind. Mat., 5, No. 2, 109–125 (2002).
7. Kononenko L. I. and Volokitin E. P., “Parameterization and qualitative analysis of a singular system in a mathematical model of reaction of catalytic oxidation [in Russian],” Sib. Zh. Ind. Mat., 15, No. 1, 44–52 (2012).
8. Voropaeva N. V. and Sobolev V. A., Geometric Decomposition of Singularly Perturbed Systems [in Russian], Fizmatlit, Moscow (2009).
9. Tikhonov A. N., “On independence of solutions to differential equations on a small parameter [in Russian],” Mat. Sb., 22, No. 2, 193–204 (1948).
10. Zvonkin A. K. and Shubin M. A., “Nonstandard analysis and singular perturbations of ordinary differential equations [im Russian],” Uspekhi Mat. Nauk, 39, No. 2, 77–127 (1984).
11. Arnold V. I. et al., “Theory of bifurcations,” in: Modern Problems of Mathematics, Fundamental Directions [in Russian], pp. 5–218, VINITI, Moscow (1986).
12. Chumakov G. A. and Chumakova N. A., “Relaxation oscillations in a kinetic model of catalytic hydrogen oxidation involving a chase on canards [in Russian],” Chem. Eng. J., 91, No. 2-3, 151–158 (2003).
13. Sobolev V. A. and Shchepakina E. A., “Duck-trajectories in one problem of combustion theory [in Russian],” Differ. Equ., 32, No. 9, 1175–1184 (1996).
14. Kononenko L. I., “Relaxations in singularly perturbed systems on the plane [in Russian],” Vestn. Novisib. Gos. Univ., Ser. Mat., Mekh., Inform., 9, No. 4, 45–50 (2009).
15. Reshetnyak Yu. G., Course of Mathematical Analysis [in Russian], Izdat. Inst. Mat., Novosibirsk, vol. 1: 1999; vol. 2.: 2000.
16. Grau I. A. and Grau M., “A survey on the unverse integrating factor,” Qual. Theory Dyn. Syst., 9, 115–166 (2010).
Review
For citations:
Kononenko L.I., Volokitin E.P. Qualitative analysis of one singularly perturbed system of differential equations with a small parameter. Mathematical notes of NEFU. 2024;31(3):15-27. (In Russ.) https://doi.org/10.25587/2411-9326-2024-3-15-27
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