Dynamics of an Inverting Tippe Top

Abstract

Abstract. The existing results about inversion of a tippe top (TT) establish stability of asymptotic solutions and prove inversion by using the LaSalle theorem. Dynamical behaviour of inverting solutions has only been explored numerically and with the use of certain perturbation techniques. The aim of this paper is to provide analytical arguments showing oscillatory behaviour of TT through the use of the main equation for the TT. The main equation describes time evolution of the inclination angle θ(t) within an effective potential V (cos θ, D(t), λ) that is deforming during the inversion. We prove here that V (cos θ, D(t), λ) has only one minimum which (if Jellett's integral is above a thresholdI3 < 1 holds) moves during the inversion from a neighbourhood of θ = 0 to a neighbourhood of θ = π. This allows us to conclude that θ(t) is an oscillatory function. Estimates for a maximal value of the oscillation period of θ(t) are given.

Dynamics of the Tippe Top—Properties of numerical solutions versus the dynamical equations

“…This proposition states that if λ > λ thres , so that TT is inverting, then the minimum z min of V(z, D(t), λ) is moving from an ǫ-neighborhood of z = 1 to an ǫ-neighborhood of z = −1. In [8] it has also been shown that the time of passage for θ(t) between two turning points has a uniform bound…”

“…A rigorous approach showing oscillatory behaviour of TT inverting solutions has been proposed in [7,8,10]. It is based on an integrated form of TT equations that are equivalent to the Euler angle equations and to the dynamical Eqs.…”

“…In [8] the deformation is considered in the special case when the effective potential V(z, D(t), λ) is a rational function of cos θ. This occurs when the physical parameters satisfy 1 − α 2 < γ < 1, σ = 1−α γ+α 2 −1 , conditions that may be realized by parameters of a real toy TT.…”