Dynamics of an Inverting Tippe Top

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 $\theta(t)$ within an effective potential $V(\cos\theta,D(t),\lambda)$ that is deforming during the inversion. We prove here that $V(\cos\theta,D(t),\lambda)$ has only one minimum which (if Jellett's integral is above a threshold value $\lambda>\lambda_{\text{thres}}=\frac{\sqrt{mgR^3I_3\alpha}(1+\alpha)^2}{\sqrt{1+\alpha-\gamma}}$ and $1-\alpha^2<\gamma=\frac{I_1}{I_3}<1$ holds) moves during the inversion from a neighbourhood of $\theta=0$ to a neighbourhood of $\theta=\pi$. This allows us to conclude that $\theta(t)$ is an oscillatory function. Estimates for a maximal value of the oscillation period of $\theta(t)$ are given.