We report extensive numerical studies of the dynamics of a classical particle in an anharmonic one-dimensional potential while it is harmonically coupled to three different many-particle systems. The studies address the comparatively simpler dynamical problem when the energy of the anharmonic oscillator is sufficiently low. The first model is one in which the many-particle system is a chain of harmonic oscillators that have nearest-neighbor harmonic interactions. The anharmonic oscillator is connected to every harmonic oscillator via harmonic springs. The second model is identical to the first except that each oscillator is subjected to an additional one-body harmonic potential. The third model is identical to the second except that the nearest-neighbor couplings between the individual harmonic oscillators are absent, i.e., the oscillators are decoupled with respect to one another. In the first model we find that the lowest frequency of the anharmonic oscillator, while possessing a lower bound, increases as $\ensuremath{\omega}\ensuremath{\propto}\sqrt{\mathrm{NK}},$ where N is the number of oscillators in the bath and K is the force constant describing the strength of interaction between the anharmonic oscillator and the interacting harmonic many-body system. In the second model we find aspects of dynamical correlations as seen in the canonical ensemble and a dominant frequency that increases as NK. We recover the dynamical correlations of the anharmonic oscillator as obtained within the framework of a canonical ensemble [S. Sen, R. S. Sinkovits, and S. Chakravarti, Phys. Rev. Lett. 77, 4855 (1996)] in the third model for sufficiently many harmonic oscillators. We comment briefly on an alternate way to correctly model heat baths for numerical studies on the dynamics of physical systems using canonical ensembles.