MECHANICAL EXCITATION TO EXTEND FLYING TIME OF LEVITRON

The Levitron is a magnetic spinning top which levitate by the interaction of an external axial magnetic field. The dynamic of this device had been studied by M. Berry, Dulling & Meiss, Gans, etc. Using Maxwell equation and rigid body equations, it is possible to obtain a six degrees of freedom Hamiltonian system.  The Levitron has stable levitation if the spinning frequency belong to a specific interval of frequencies. We include dissipation for a more realistic model of the Levitron, but we introduce two types of mechanical forcing (parametric resonance and a second static magnet) to inject energy into the system in order to prevent the prompt falling of the spinnig top as well. An asymptotic multiscale analysis is also carried out with the aim of studing the nonlinear interaction between the traslational and rotational modes. A systematic study of the flying time as a function of the perturbation parameters is performed, and detailed bifurcation diagrams are obtained exhibiting an Arnolds's tongues structure. Avery 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.