Dynamics of the Tippe Top -- properties of numerical solutions versus the dynamical equations

We study the relationship between numerical solutions for inverting Tippe Top and the structure of the dynamical equations. The numerical solutions confirm oscillatory behaviour of the inclination angle $\theta(t)$ for the symmetry axis of the Tippe Top. They also reveal further fine features of the dynamics of inverting solutions defining the time of inversion. These features are partially understood on the basis of the underlying dynamical equations

Dynamics of the Tippe Top via Routhian Reduction

We consider a tippe top modeled as an eccentric sphere, spinning on a horizontal table and subject to a sliding friction. Ignoring translational effects, we show that the system is reducible using a Routhian reduction technique. The reduced system is a two dimensional system of second order differential equations, that allows an elegant and compact way to retrieve the classification of tippe tops in six groups as proposed in [1] according to the existence and stability type of the steady states.Comment: 16 pages, 7 figures, added reference. Typos corrected and a forgotten term in de linearized system is adde

On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.Comment: 44 pages, introduction revised, references expanded. To appear in Arch. Rational Mech. Ana

Dynamics of a rolling and sliding disk in a plane. Asymptotic solutions, stability and numerical simulations

We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straight-line rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoopFunding agencies:  National Science Center of Poland [DEC-2013/09/B/ST1/04130]; Department of Mathematics of Linkoping University</p