On periodic motions of a hamiltonian system with two degrees of freedom in the vicinity of a multiple resonance of the third order

DOI: 10.34759/trd-2022-126-02

Аuthors

Safonov A. I.

InfoSistem-35, 16, 3-Mytischinskaya str., bld. 37, Moscow, 129626, Russia

e-mail: lexafonov@mail.ru

Abstract

In this paper, we study the motion of a non-autonomous 2π-periodic in time Hamiltonian system with two degrees of freedom in a neighborhood of a trivial equilibrium that is stable in the linear approximation. It is assumed that the system contains a small parameter ε, and for ε = 0 the Hamiltonian of the system does not depend on time. Let the values of the parameters be close to the resonance values corresponding to the double (fundamental and combination) resonance of the third order. Then, the trivial equilibrium of the complete system is unstable. It is assumed that there is a resonant detuning in one of the frequencies of the linear oscillations of the system.

The goal of the paper is to solve the question of the existence, number and stability (in the linear approximation) of periodic motions of the system in a small neighborhood of the origin. Using a number of canonical transformations, the Hamiltonian functions are reduced to the forms that are characteristic for each resonance case. Model systems corresponding to autonomous systems are studied. The parameter space of the problem is divided into regions, and in each region the question of the existence and number of resonance equilibrium positions of the model systems is solved. The results are compared with ones in the case of exact resonance.

Using the Sylvester criterion, the sufficient conditions for the stability of the equilibrium positions are verified, and the equilibrium positions satisfying them are found. The characteristic equation of the linearized system of equations of perturbed motion is analyzed, and the necessary conditions for stability of the equilibrium positions are obtained in the form of the system of inequalities The equilibrium positions are found, for which one of these conditions is satisfied; the remaining equilibrium positions are unstable (in the complete model systems) A complete analysis of the necessary conditions for stability has not been carried out due to cumbersomeness.

Using the Poincare small parameter method, the periodic motions generated by the considered equilibrium positions are constructed in the complete non-autonomous systems. They are analytic in ε and 12π-periodic in t. The conclusions are drawn about their stability (in the linear approximation) or instability.

Keywords:

Hamiltonian system, double third-order resonance, stability in the linear approximation, periodic motions

References

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