Towards a Prototype of a Spherical Tippe Top

Figure 14

Trajectories of the system (3.15)-(3.16) projected onto the subspace of variables (𝛼,𝜃,𝑛) for an eccentric ball of Group IIa with friction 𝜇=0.04. The bold curves are the intermediate states (5.10). The curves 𝑠₁, 𝑠₂, and 𝑠₃ in (𝛼,𝑛)-plane represent the curves on which the initial value of 𝛼 is chosen. The starting angle is 𝜃=𝜋−0.01. The arrows indicate the final position. Points on the plane 𝜃=𝜋 are stable if |𝑛|>𝑛₊, unstable otherwise. Trajectories are attracted to a noninverted position on the plane 𝜃=0 for |𝑛|<𝑛_𝑏, or they reach a point on the red curves otherwise. Two trajectories are added to illustrate the initial wild oscillations when 𝛼 is not chosen on one of the 𝑠_𝑖 curves. We used 𝑚=𝑔=1, 𝜖/𝑅=0.5, 𝐴/𝐶=0.55, and 𝜇=0.04.