Nonlinear oscillations of sympathetic pendulums

Nonlinear problem of motion of two identical pendulums connected by an elastic spring in the neighborhood of their stable vertical equilibrium is investigated. Stiffness of the spring is supposed small, i.e. the case close to resonance $1:1$ is considered. The problem of existence and orbital stability of periodical motions of the pendulums arising from the equilibrium is solved. It is indicated existence of motions asymptotic to one of the periodical motions. An analysis of quasi-periodical motions of an approximate system is given in which members up to the forth order inclusively in the normalizing Hamiltonian of the problem are taken into account. Using KAM-theory the question is considered of preservation of these motions in the complete nonlinear system in which members of all orders in the series expansion of Hamiltonian in the sufficiently small neighborhood of the equilibrium are taken account.