The non-linear oscillations of a pendulum of variable length on a vibrating base

A generalized scheme for averaging a system with several small independent parameters is described: equations of the first and second approximations are obtained, and an estimate is made of the accuracy of the approximation and the value of the asymptotically long time interval. The problem of the oscillations of a pendulum of variable length on a vibrating base for high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and its suspension point is considered. Averaged equations of the first and second approximations are obtained, and the bifurcations of the steady motions in the equations of the first approximation, and also in the second approximation for 1:2 resonance, are obtained. One of the possible bifurcations of the phase portrait in the neighbourhood of 1:2 resonance is described based on a numerical investigation. It is shown that a change in the resonance detuning parameter from zero to a value of the first order of infinitesimals in the small parameter leads to stabilization of the upper equilibrium position through a splitting of the separatrices for the resonance case; the splitting of separatrices is accompanied by the occurrence of a stochastic web in the neighbourhood of this equilibrium, its localization, and subsequent contraction to an equilibrium point and the formation of a new oscillation zone.