Mechanical stabilization of the dissipative model for the Levitron: Bifurcation study and early prediction of flight times

Abstract

We numerically integrate the equations of motion of the Levitron in its twofold fashion, i.e. in terms of the Eulerian description of the spinning top’s motion as well as those in a different set of angular coordinates, the yaw-pitch-roll angles, in order to avoid the singularity posed by the vanishing of the angle describing the top’s nutation. We not only extend both set of equations to include dissipation for a more realistic model of the Levitron, but we introduce two types of mechanical forcing to inject energy into the system to prevent the prompt falling of the spinning top as well. A systematic study of the flying time as a function of the perturbation parameters is performed, and detailed bifurcation diagrams are obtained exhibiting an Arnold’s tongues structure. A very similar structure is obtained when the stability analysis is carried out by recourse to a fast method to compute the maximum Lyapunov exponent, namely the Mean Exponential Growth factor of Nearby Orbits (MEGNO). Our numerical experiments confirmed that the MEGNO serves as an early indicator of the stability of the Levitron’s flights, regular solutions being good candidates to allow for very long flying times.