Rus. J. Nonlin. Dyn., 2025 Volume 21, Number 4,Pages 623–648(Mi nd974)
In Memory of Alexey V. Borisov. On His 60th Birthday
On the Motions of a Nearly Autonomous Hamiltonian System at Parameter Values Close to the Boundary of Stability Regions of the Limiting Autonomous Problem
Abstract:
The paper studies the motions of a nearly autonomous, Hamiltonian system $2\pi$-periodic
in time, with two degrees of freedom, in the neighborhood of a trivial equilibrium. The values
of the parameters are considered near the boundary of the stability region of this equilibrium
that corresponds to the zero frequency case in the limiting autonomous problem. The cases
are distinguished when the other frequency is nonresonant (not equal to an integer or half-integer number) and resonant, i. e., a multiple parametric resonance is realized in the system.
The stability and instability regions (parametric resonance regions) of the trivial equilibrium
of the system are obtained. The question of the existence in its neighborhood of analytic (in
integer or fractional powers of a small parameter) resonant periodic motions, their number and
linear stability is resolved. Earlier, similar results for the studied cases of multiple parametric
resonances have been obtained in the sections of the parameter space corresponding to a fixed
(resonant) value of one of the parameters. In this paper, complete neighborhoods of resonance
points in the parameter space are considered. As expected, this leads to complication of both
the stability diagram of the trivial equilibrium and the distribution and character of the stability
of the periodic solutions under study. For each resonance case, an algorithm for analyzing these
motions is developed. Differences and similarities in the properties of the system for different
resonant cases are established, parallels are found with multiple resonance cases of another type,
arising in the systems with another structure of the perturbing part of the Hamiltonian function.
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