Numerical stabilization of the Levitron's realistic model

Abstract

The stability of the magnetic levitation showed by the Levit- 10 ron was studied by M.V. Berry as a six degrees of freedom Hamiltonian 11 system using an adiabatic approximation. Further, H.R. Dullin found 12 critical spin rate bounds where the levitation persist and R.F. Gans 13 et al. offered numerical results regarding the initial conditions’ manifold 14 where this occurs. In the line of this series of works, first, we extend the 15 equations of motion to include dissipation for a more realistic model, 16 and then introduce a mechanical forcing to inject energy into the sys- 17 tem in order to prevent the Levitron from falling. A systematic study 18 of the flying time as a function of the forcing parameters is carried out 19 which yields detailed bifurcation diagrams showing an Arnold’s tongues 20 structure. The stability of these solutions were studied with the help 21 of a novel method to compute the maximum Lyapunov exponent called 22 MEGNO. The bifurcation diagrams for MEGNO reproduce the same 23 Arnold’s tongue structure.