The inverse-square relationship of universal gravitation is crucial for the stability, especially the closedness, of planetary orbits. This paper studies the closedness of a trivial circular orbit of a mass point in a special non-gravitational central potential field (spring elastic potential energy) after being perturbed. In the gravitational field, the orbits of bounded motion of mass points are always closed. However, in this non-gravitational central potential field, both radial and tangential velocity perturbations will destroy the closedness of the original circular orbit and cause oscillations in the rotation period of the mass point around the center. Over a long period of time, the trajectory of the mass point fills the space region in the motion plane allowed by energy conservation. Under specific parameters, the trajectory of the mass point is sensitive to initial perturbation conditions and exhibits partial characteristics of chaos. These research contents, on the one hand, can help students master and apply the conservation relationships of energy and angular momentum skillfully, and on the other hand, can deepen the understanding of the motion of mass points in central potential fields.