  The system in which a small rigid ball is bouncing repeatedly on a massive at
table vibrating vertically, so-called the bouncing ball system, has been widely
studied. Under the assumption that the table is vibrating with a piecewise
polynomial function of time, the bifurcation diagram changes qualitatively
depending on the order of the polynomial function. We elucidate the mechanism
of the difference in the bifurcation diagrams by focusing on the two-period
solution. In addition, we derive the approximate curve of the branch close to
the period-doubling bifurcation point in the case of the piecewise cubic
function of time for the table vibration. We also performed numerical
calculation, and we demonstrate that the approximations well reproduce the
numerical results.
