Journal of Statistical Physics, Vol. 60, Nos. 1/2, 1990. p. 245-262.
We analyze a large system of limit-cycle oscillators with mean-field coupling and randomly distributed natural frequencies. We prove that when the coupling is sufficiently strong and the distribution of frequencies has sufficiently large variance, the system undergoes "amplitude death" — the oscillators pull each other off their limit cycles and into the origin, which in this case is a stable equilibrium point for the coupled system. We determine the region in coupling-variance space for which amplitude death is stable, and present the first proof
that the infinite system provides an accurate picture of amplitude death in the large but finite system.