When b50, it is straightforward to show that v52c, which
means that the ball undergoes simple rolling at the point of
contact after the first bounce. Thus for 21,b,1, Eq. ~15!
indicates that the ball undergoes a transition to simple rolling
as n→`. Figure 2~b! shows this transition for the initial
conditions used in Fig. 2~a!. This transition captures an aspect
of superball motion that many find very useful; it is
much easier to pick up a dropped ball if you wait until the
erratic bouncing decays to simple rolling.
Finally, consider the eigenvalues and eigenvectors of M.
The first eigenvalue for this matrix is l1511 and the corresponding
eigenvector is