Geodesic
Methods for studying the stability of linear periodic systems depending on a small parameter
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 113-126.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider systems of linear differential equations with periodic coefficients depending on a small parameter. We propose new approaches to the problem of constructing a monodromy matrix that lead to new effective formulas for calculating multipliers of the system studies. We present a number of applications in problems of the perturbation theory of linear operators, in the analysis of stability of linear differential equations with periodic coefficients, in the problem of constructing the stability domains of linear dynamical systems, etc.
Keywords: differential equation, periodic system, Hamiltonian system, stability, small parameter.
Mots-clés : monodromy matrix, multiplier 
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M. G. Yumagulov; L. S. Ibragimova; A. S. Belova. Methods for studying the stability of linear periodic systems depending on a small parameter. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 113-126. http://geodesic.mathdoc.fr/item/INTO_2019_163_a8/

[1] Demidenko G. V., Matveeva I. I., “Ob ustoichivosti reshenii lineinykh sistem s periodicheskimi koeffitsientami”, Sib. mat. zh., 42:2 (2001), 332–348 | MR | Zbl

[2] Ibragimova L. S., Mustafina I. Zh., Yumagulov M. G., “Asimptoticheskie formuly v zadache postroeniya oblastei giperbolichnosti i ustoichivosti dinamicheskikh sistem”, Ufim. mat. zh., 2016, no. 3, 59–81 | MR

[3] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1975

[4] Krasnoselskii M. A., Yumagulov M. G., “Metod funktsionalizatsii parametra v probleme sobstvennykh znachenii”, Dokl. RAN., 365:2 (1999), 162–164 | MR

[5] Malkin I. G., Metody Lyapunova i Puankare v teorii nelineinykh kolebanii, Editorial URSS, M., 2004 | MR

[6] Perov A. I., “Ob odnom kriterii ustoichivosti lineinykh sistem differentsialnykh uravnenii s periodicheskimi koeffitsientami”, Avtomat. telemekh., 2013, no. 2, 22–37 | Zbl

[7] Rozo M., Nelineinye kolebaniya i teoriya ustoichivosti, Nauka, M., 1971

[8] Chezari L., Asimptoticheskoe povedenie i ustoichivost reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1964

[9] Yumagulov M. G., Ibragimova L. S., Imangulova E. S., “Glavnye asimptotiki v zadache o bifurkatsii Andronova—Khopfa i ikh prilozheniya”, Differ. uravn., 53:12 (2017), 1627–1642 | DOI | MR | Zbl

[10] Yumagulov M. G., Ibragimova L. S., Mustafina I. Zh., “Issledovanie granits oblastei ustoichivosti dvukhparametricheskikh dinamicheskikh sistem”, Avtomat. telemekh., 2017, no. 10, 74–89 | Zbl

[11] Yakubovich V. A., Starzhinskii V. M., Lineinye differentsialnye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya, Nauka, M., 1972 | MR

[12] Chiang H. D., Alberto L. F., Stability Regions of Nonlinear Dynamical Systems: Theory, Estimation, and Applications, Cambridge Univ. Press, Cambridge, 2015 | MR | Zbl

[13] Jianjun Paul Tiana, Jin Wangb, “Some results in Floquet theory, with application to periodic epidemic models”, Appl. Anal., 94:6 (2015), 1128–1152 | DOI | MR | Zbl

[14] Seyranian A. P., Mailybaev A. A., Multiparameter Stability Theory with Mechanical Applications, World Scientific, New Jersey, 2003 | MR | Zbl

[15] Traversa F. L., Di Ventra M., Bonani F., “Generalized Floquet theory: application to dynamical systems with memory and Bloch's theorem for nonlocal potentials”, Phys. Rev. Lett., 110 (2013), 170602 | DOI