Oscillations of a Coupled Controlled System near Equilibrium

Abstract

A nonlinear autonomous system near an equilibrium is considered. The matrix of its linearized counterpart is supposed to have imaginary eigenvalues without internal resonances up to the fourth order inclusive. The oscillations of this system caused by periodic controls with a small gain k are investigated, and isolated resonant oscillations are found. The amplitudes of the oscillations in terms of the parameter k are estimated, and their stability is analyzed. It is shown that the existence of a resonant oscillation is guaranteed by the control action, while its asymptotic stability is determined by the uncontrolled system.