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The stability of Hamiltonian systems in the case of a multiple fourth- order resonance. (English. Russian original) Zbl 0793.58017

J. Appl. Math. Mech. 56, No. 4, 572-576 (1992); translation from Prikl. Mat. Mekh. 56, No. 4, 672-675 (1992).
2-dimensional Hamiltonian systems are considered. It is assumed that the Hamiltonian is not definite and that the eigenvalues of the system are pure imaginary and different. Two interesting cases are studied. In the first one the system has two independent fourth order resonances, in the second it has two fourth order resonances coupled through one eigenvalue. Several theorems are proved that provide conditions of stability in both cases.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37A30 Ergodic theorems, spectral theory, Markov operators
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[1] Kunitsyn, A. L.; Markeyev, A. P., Itogi Nauki i Tekhniki, Obshchaya Mekhanika, (Stability in Resonant Cases, Vol. 4 (1979), VINITI: VINITI Moscow)
[2] Birkhof, D., Dynamic Systems (1941), Gostekhizdat: Gostekhizdat Moscow and Leningrad
[3] Markeyev, A. P., On the stability of a canonical system with two degrees of freedom, in the presence of a resonance, Prikl. Mat. Mekh., 32, 738-744 (1968) · Zbl 0184.11904
[4] Kunitsyn, A. L.; Medvedev, S. V., On stability in the presence of several resonances, Prikl. Mat. Mekh., 41, 422-429 (1977)
[5] Kunitsyn, A. L.; Perezhogin, A. A., The stability of neutral systems in the case of a multiple fourth-order resonance, Prikl. Mat. Mekh., 49, 72-77 (1985) · Zbl 0594.70026
[6] Khazin, L. G., On the stability of Hamiltonian systems in the presence of a resonance, Prikl. Mat. Mekh., 35, 423-431 (1971)
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