Numerical stabilization of the Levitron's realistic model

CLAUDIA GIORDANO; ABRAHAM DE LA ROSA ; ARTURO OLVERÁ

EUROPEAN JOURNAL OF PHYSICS

IOP PUBLISHING LTD

Lugar: Londres; Año: 2016 vol. 225 p. 2729 - 2740

0143-0807

The stability of the magnetic levitation showed by the Levitron wasstudied by M. V. Berry as a six degrees of freedom Hamiltonian system using an adiabatic approximation. Further, H. R. Dullin found  critical spin rate bounds where the levitation persist and R. F. Gans et al. offered numerical results regarding the initial conditions´ manifold where this occurs.  In the line of this series of works, first, we extend the equations of motion to include dissipation for a more realistic model, and then introduce two different kinds of mechanical forcing to inject energy into the system in order to prevent the Levitron from falling. We perform a numerical study of the complete system finding bounds for the tilt and spining rate of the top. A systematic study of the flying time as a function of  the forcing parameters is carried out which  yieldsdetailed bifurcation diagrams showing an Arnold´s tongues structure.  The stability of these solutions were studied with the help of a novel method to compute the maximum Lyapunov exponent called MEGNO. The bifurcation diagrams for MEGNO reproduce the same Arnold´s tongue structure.