result implies an $N$-body system with number of particles larger than a critical number should show the gravothermal oscillation.

Since several other calculations based on similar models did not reproduce the result of Sugimoto and Bettwieser (1983), the validity of their result had been controversial for a few years after their first paper. However, calculations with improved accuracy confirmed that the oscillation takes place in both the gaseous models and Fokker-Planck (FP) calculations([@Goodman1987], [@Heggie1989], [@Cohn1989]).

What have been found in gas models and FP calculations are summarized as follows. There is a homologously expanding solution for the post-collapse evolution of the cluster ([@Goodman1984]). In this solution, the size of the core is proportional to $N^{-1/3}$. Thus, the core is smaller for larger number of particles. If the core is too small, this homologous expansion becomes unstable in a way similar to the way in which an isothermal sphere is unstable, and the core starts to show an oscillating behavior. The critical number of particles is around 7,000 (Goodman 1987). If the total number of particle is slightly larger than this critical value, the oscillation is regular. However, at $N\simeq 10,000$, “period doubling” takes place and the density shows oscillation with 