Stability of coupled oscillators

We study a system of two coupled oscillators and a modified system of these oscillators whose rods intersect and slide without friction relative to each other. The oscillators posed vertically in a uniform gravity field and its interaction is described by a potential depending on distance. We demonstrate that both systems have symmetrical and asymmetrical equilibrium states. Stability of the states depend on the interaction energy and distance between the oscillators' suspension centers. Stability regions for Hooke and Coulomb potentials are calculated in the parameter plane.