On the problem of the stability of a Hamiltonian system with one degree of freedom on the boundaries of regions of parametric resonance

Abstract

A one–degree–of–freedom system that is periodic in time is considered in the vicinity of its equilibrium position in the case of multiple multipliers of the linearized system. It is assumed that the monodromy matrix is reduced to diagonal form and, therefore, the equilibrium is stable in a first approximation. An algorithm for constructing a canonical transformation that brings the system into such a form, in which the terms of high (finite) order are eliminated in the expansion of the Hamiltonian into a time series and the second-order terms are totally absent, is described. The stability and instability conditions are found using Lyapunov's second method and KAM (Kolmogorov–Arnold–Moser) theory in one particular case, in which the stability problem is not solvable for the third- and fourth-order forms in the expansion of the original Hamiltonian into a series.