RESONANCES AND ASYMPTOTIC TRAJECTORIES IN HAMILTONIAN-SYSTEMS

The existence of motions asymptotic to the equilibrium state of a Hamiltonian system with an arbitrary finite number of degrees of freedom is investigated. It is assumed that the Hamiltonian function is analytical in the neighbourhood of the equilibrium and is either time-periodic or time-independent. The characteristic exponents of the linearized equations of motion are purely imaginary and a simple third- or fourth-order resonance is observed. The sufficient conditions for asymptotic motions to exist are derived, and their approximate analytical representation is constructed in a fairly small neighbourhood of the position of equilibrium.