Abstract:
This is a study of a dynamical system depending on a parameter κ. Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on κ, the focus is on details of stability loss through various bifurcations (Poincaré–Andronov–Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, κ is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, κ varies slowly with time (the case of a dynamic bifurcation). In the simplest situation κ=εt, where ε is a small parameter. More generally, κ(t) may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay.
Bibliography: 88 titles.
Keywords:
Lyapunov stability, bifurcation of an equilibrium, bifurcation of a periodic solution, soft stability loss, hard stability loss, stability loss delay.
The research of the second author (§ § 2 and 3) was supported by the Russian Science Foundation under grant no. 20-11-20141 and performed in the Steklov Mathematical Institute of Russian Academy of Sciences.
Citation:
A. I. Neishtadt, D. V. Treschev, “Dynamical phenomena connected with stability loss of equilibria and periodic trajectories”, Russian Math. Surveys, 76:5 (2021), 883–926
\Bibitem{NeiTre21}
\by A.~I.~Neishtadt, D.~V.~Treschev
\paper Dynamical phenomena connected with stability loss of equilibria and periodic trajectories
\jour Russian Math. Surveys
\yr 2021
\vol 76
\issue 5
\pages 883--926
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Linking options:
https://www.mathnet.ru/eng/rm10023
https://doi.org/10.1070/RM10023
https://www.mathnet.ru/eng/rm/v76/i5/p147
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O. S. Kipkaeva, E. A. Shchepakina, “STABILITY CHANGE OF INVARIANT MANIFOLDS OF DIFFERENTIAL SYSTEMS WITH MULTI-SCALE VARIABLES”, Differencialʹnye uravneniâ, 60:9 (2024)
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D. V. Treschev, E. I. Kugushev, T. V. Popova, S. V. Bolotin, Yu. F. Golubev, V. A. Samsonov, Yu. D. Selyutskii, “Kafedra teoreticheskoi mekhaniki i mekhatroniki”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2024, no. 6, 103–113
D. V. Treschev, E. I. Kugushev, T. V. Shakhova, S. V. Bolotin, Yu. F. Golubev, V. A. Samsonov, Yu. D. Selyutskiy, “Chair of Theoretical Mechanics and Mechatronics”, Moscow Univ. Mech. Bull., 79:6 (2024), 200
D. Bigoni, F. Dal Corso, O. N. Kirillov, D. Misseroni, G. Noselli, A. Piccolroaz, “Flutter instability in solids and structures, with a view on biomechanics and metamaterials”, Proc. R. Soc. A, 479:2279 (2023)
J. Wen, H. Zhang, Zh. Wu, Q. Wang, H. Yu, W. Sun, B. Liang, Ch. He, K. Xiong, Y. Pan, Y. Zhang, Zh. Liu, “All-optical spiking neural network and optical spike-time-dependent plasticity based on the self-pulsing effect within a micro-ring resonator”, Appl. Opt., 62:20 (2023), 5459
A. Bazzani, F. Capoani, M. Giovannozzi, “Analysis of adiabatic trapping phenomena for quasi-integrable area-preserving maps in the presence of time-dependent exciters”, Phys. Rev. E, 106:3 (2022), 034204, 13 pp.
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