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Semi-invariant form of equilibrium stability criteria in critical cases. (English. Russian original) Zbl 0632.58028

J. Appl. Math. Mech. 50, 543-546 (1986); translation from Prikl. Mat. Mekh. 50, 707-711 (1986).
Expressions are given for the first coefficients of the normal form of the equations in the neutral equilibrium manifolds of an autonomous system in the basic critical cases, in terms of the neutral eigenvectors of the linearized system and its conjugate. Using the reduction principle [V. A. Pliss, Izv. Akad. Nauk SSSR, Ser. Mat. 28, 1297-1324 (1964; Zbl 0131.315)], the stability criteria previously obtained by various authors in critical cases take an explicit form, which is very convenient for calculations and does not involve the restriction of finite dimensionality.

MSC:

37C10 Dynamics induced by flows and semiflows

Citations:

Zbl 0131.315
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References:

[1] Pliss, V.A., The reduction principle in the theory of stability of motion, (), 6 · Zbl 0134.30701
[2] Lyapunov, A.M., The general problem of the stability of motion, Collected papers, izd- vo akad. nauk SSSR Moscow-leningrad, 2, (1956) · Zbl 0041.32204
[3] Lyapunov, A.M., Study of a singular case of the problem of stability of motion, Collected papers izd-vo akad. nauk SSSR, Moscow-leningrad, 2, (1956)
[4] Khazin, L.G.; Shnol’, E.E., Stability of equilibrium positions in critical cases and close-to-critical cases, Pmm, 45, 4, (1981) · Zbl 0511.34036
[5] Marsden, D.; McCracken, M., Bifurcation of the birth of a cycle and applications, (1980), Mir Moscow
[6] Arnol’d, V.I., Supplementary chapters on the theory of ordinary differential equations (dopolnitel’nye glavy teorii obyknovennykh differentsial’nykh uravnenii), (1978), Nauka Moscow · Zbl 0956.34503
[7] Molchanov, A.M., Stability when the linear approximation is neutral, Dokl. akad. nauk, SSSR, 141, 1, (1961) · Zbl 0115.07501
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