On the stability of coupled nonlinear oscillators

In this paper, we consider the problem of Yu. N. Bibikov on the stability of equilibrium positions of two coupled nonlinear oscillators under the action of conservative perturbing forces. We obtain stability and instability conditions for the case of sufficiently small perturbing forces. Also, we consider the problem of stabilizing an equilibrium position by potential forces when only the relative position of the oscillators is measured and propose the form of a stabilizing potential.