with number of particles 2,048–32,768. This is the first calculation with the number of particles well beyond the critical number of 7,000 which covers sufficiently long time after the collapse to determine the nature of the post-collapse evolution. All simulations were performed on GRAPE-4 ([@Taiji1996]), a special-purpose computer for collisional $N$-body simulations. Our main results are the following. First, for large $N$, the core density and core mass exhibited oscillations with large amplitude. The core mass at the maximum expansion is in good agreement with FP or gas model results (1% of the total mass for 16k- and 32k-particle runs). Second, we confirmed that the observed oscillation is driven by gravothermal instability. Several long expanding periods without any energy input were observed. The temperature inversion was visible in such expansion phases. The behavior of the core density is strikingly similar to the result of FP calculations with stochastic heat source ([@Takahashi1991]), which strongly suggests that the mechanism is the same. The trajectory in the central density-central temperature plane clearly indicates that the inward heat flux supports the expansion.

The structure of this paper is the following. In section 2, we describe the initial model, numerical method and the computer used. 