Perturbation theory methods in problem of parametric resonance for linear periodic Hamiltonian systems

Abstract

We consider the problem on parametric resonance for linear periodic Hamiltonian systems depending on a small parameter. We propose new formulae based on the methods of the perturbation theory for linear operators in the problem on approximate construction of multipliers for linear non-autonomous periodic Hamiltonian systems. We focus on obtaining the formulae for the first correctors of perturbations of multiple definite and indefinite multipliers. The proposed formulae lead to new Lyapunov stability criteria for linear periodic Hamiltonian systems in critical cases. We consider applications to the problem on parametric resonance in main resonances. The obtained results are formulated in terms of the original equations and lead us to effective formulae and algorithms. The effectiveness of the proposed formulae is demonstrated by solving the problem of plotting the boundaries of the stability regions of triangular libration points of a planar bounded elliptic three-body problem.