Abstract The coupling is established analytically between the natural oscillations of stars normal to the mean horizontal plane and periodic changes in the excitation force in a rotationally flattened galactic disk using a linear perturbation framework. The nonaxisymmetric gravitational potential of a density-wave bar may cause these changes in this ‘pumping’ force. The excitation of out-of-plane oscillating motion with an amplitude exponentially increasing with time via parametric resonance interaction between stars and a bar is considered for the formation of the boxy/peanut-shaped bulges (an X-structure of stars crossing at the center of the system) often observed in disk galaxies, even in isolation, as well as in the Galaxy. A scenario involving this resonant interaction accounts for both the vertically short 1 ≲ 1 kpc boxy/peanut-shaped features and a 1 ∼ 1 Gyr time delay between when the bars form and vertically thicken.
Griv et al. (2026) studied this question.