Stabilization of in-phase motions in symmetrically coupled oscillators by parametric perturbations

In this work we consider the possibility of using high frequency periodic parametric influence for the purpose of synchronization of oscillations. The term “synchronization” is used here in a narrow sense as motions in a symmetric subspace (x_l(t)=x₂(t)). The idea of this method is originated from a classical physical task of a pendulum with vibrating suspension. As it was shown in the literature that certain values of the amplitude and the frequency of the vibration lead to stability of the upper equilibrium of the pendulum. It is known that the task of stability of in-phase motions can be reduced to stability of the zero fixed point of some systems. As high frequency parametric excitation can change the stability of an equilibrium, we can suppose that it can also change the stability of the symmetric motions. Coupled identical oscillators forced in-phase was modelled to be used for testing the described method of synchronization