By using numerical and semiclassical methods, we evaluate the quantum breaking, or Ehrenfest time for a wave packet localized around classical equilibrium points of autonomous one-dimensional systems with polynomial potentials. We find that the Ehrenfest time diverges logarithmically with the inverse of the Planck constant whenever the equilibrium point is exponentially unstable. For stable equilibrium points, we have a power law divergence with exponent determined by the degree of the potential near the equilibrium point.

4 pages, 5 figures