On the stability of periodic motions of an autonomous Hamiltonian system in a critical case of the fourth-order resonanceстатья
Статья опубликована в высокорейтинговом журнале
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Аннотация:The problem of orbital stability of a periodic motion of an autonomous two-degree-of-
freedom Hamiltonian system is studied. The linearized equations of perturbed motion always
have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure
of the system. The other two multipliers are assumed to be complex conjugate numbers with
absolute values equal to one, and the system has no resonances up to third order inclusive, but
has a fourth-order resonance. It is believed that this case is the critical one for the resonance,
when the solution of the stability problem requires considering terms higher than the fourth
degree in the series expansion of the Hamiltonian of the perturbed motion.
Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability
are obtained, which are represented in the form of inequalities depending on the coefficients of
series expansion of the Hamiltonian up to the sixth degree inclusive.