Chaotic Motion of a Magnet Levitated Over a Superconductor

For a permanent magnet (PM) levitated over a high-temperature superconductor (HTSC, assumed to be fixed), unavoidable disturbances make it vibrate freely, and then, its complex dynamic stability due to the nonlinear hysteresis interaction between both of them rises to be a key issue. For even the simplest levitation system consisting of only one PM and one HTSC, the complex dynamic behaviors such as the period doubling bifurcation and the quasi-periodic and chaotic vibration were observed very early, but few complete theoretical studies have been reported so far. The intrinsic nonlinearity of the hysteretic interaction between the HTSC and PM, which comes from strong flux pinning in the HTSC, is believed to be the main cause of these complex dynamic behaviors. For such a nonlinear levitation system, deep understanding of the long-term anomalous dynamic behavior, particularly the chaotic motion, of the levitated body (usually PM) resulted from the nonlinear hysteresis interaction has to be very significant. In this paper, based on the work presented by Hikihara et al., we proposed a modified model, and for its mathematical equations, we revised the term of describing maximum levitation force that enables us to obtain an accurate prediction of the effect of the moving speed of the PM on the hysteresis relation of the levitation force with the distance between the PM and the HTSC. On the modified model, we studied the two routes from the period doubling bifurcation and quasi-periodic vibration to chaos and obtained the bifurcation diagram, Poincare map, the power spectrum (specifically the power spectrum density), and the maximum Lyapunov exponent. The numerical results indicate that the motion of the PM definitely tends to chaos under certain conditions.