FP calculations.

For runs with small $N$ (2k and 4k), there are some oscillation-like features but they are hardly distinguishable from fluctuations. This result is similar to that of previous studies ($e.g.$ Makino 1989). For large values of $N$, however, the oscillation with large amplitude is clearly visible.

Note that in figure 1 there is no clear transition from stable expansion to oscillation or from regular oscillation to chaotic oscillation, which were observed in gas and FP models ([@Heggie1989], [@Breeden1994]). The reason is that binaries emit energy intermittently ([@Makino1987]). Takahashi and Inagaki (1991) incorporated this stochastic nature of the energy source to their FP model and found that the core density shows chaotic oscillatory behavior even if the energy production rate is larger than the critical value for transition between stable expansion and regular oscillation. They also found that the amplitude of the oscillation is smaller for larger energy input (smaller $N$), which is consistent with the present result. In fact, it would be difficult to distinguish between our $N$-body calculation results from their stochastic FP results. Only visible difference is that their result is smoother while the central density is low. This is because the FP result is perfectly 