On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance

The points of suspension of two identical pendulums moving in a homogeneous gravitational field are located on a horizontal beam performing harmonic oscillations of small amplitude along a fixed horizontal straight line passing through the points of suspension of the pendulums. The pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the frequency of external periodic action on the system is equal simultaneously to two its frequencies of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and stability of periodic motions of pendulums with a period equal to the period of oscillations of the beam. The study uses the classical methods due to Lyapunov and Poincaré, KAM (Kolmogorov, Arnold and Moser) theory, and algorithms of computer algebra.
The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation).