  Energy level statistics of a bounded quantum system, whose classical
dynamical system exhibits bifurcations, is investigated using the two-point
correlation function (TPCL), which at the bifurcation points exhibits periodic
spike oscillations owing to the accumulation of levels called the shell effect.
The spike oscillations of the TPCL is analyzed by the reduced chi-squared value
which deduced to exhibit abrupt increases at bifurcation points, thereby
yielding a novel detection approach. Using this method, we attempt to
numerically detect the bifurcation points of a lemon-shaped billiard.
