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On the stability of a periodic Hamiltonian system with one degree of freedom in a transcendental case. (English. Russian original) Zbl 1436.70006
Dokl. Math. 97, No. 2, 161-163 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 479, No. 5, 485-488 (2018).
Summary: The stability of an equilibrium of a nonautonomous Hamiltonian system with one degree of freedom whose Hamiltonian function depends \(2\pi\)-periodically on time and is analytic near the equilibrium is considered. The multipliers of the system linearized around the equilibrium are assumed to be multiple and equal to 1 or \(-1\). Sufficient conditions are found under which a transcendental case occurs, i.e., stability cannot be determined by analyzing the finite-power terms in the series expansion of the Hamiltonian about the equilibrium. The equilibrium is proved to be unstable in the transcendental case.

MSC:
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
34D20 Stability of solutions to ordinary differential equations
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