How to twirl a hula hoop

We consider the twirling of a hula hoop when the waist of a gymnast moves along an elliptical trajectory close to a circle. For a circular trajectory, two families of exact solutions are obtained, corresponding to twirling of the hula hoop with a constant angular speed equal to the speed of the excitation. We show that one family of solutions is stable, and the other one is unstable. These exact solutions allow us to obtain approximate solutions for a slightly elliptical trajectory of the waist. We demonstrate that to twirl a hula hoop the waist needs to rotate with a phase difference between $π/2$ and $π$⁠. An interesting effect of inverse twirling is described when the waist moves in a direction opposite to the hula hoop rotation. The approximate analytical solutions are compared with the results of a numerical calculation.