When does gravity based on a specific function result in periodic orbits?

Question

Suppose there is a planet orbiting around a massive star, such that the star pulls the planet with a force proportional to a function $g(x)$ of the distance from the star (ignore the planet's force on the star). Newtonian gravity, which corresponds to $g(x) = \frac{k}{x^2},$ guarantees that every bounded orbit is periodic. $g(x) = kx$ also guarantees this because the components of the planet's motion are simple harmonic motion with the same period. For what functions $g(x)$ is the orbit guaranteed to be periodic if it is bounded, no matter what the initial position and velocity of the planet are?