Are long-term $N$-body simulations reliable?
  $N$-body integrations are used to model a wide range of astrophysical
dynamics, but they suffer from errors which make their orbits diverge
exponentially in time from the correct orbits. Over long time-scales, their
reliability needs to be established. We address this reliability by running a
three-body planetary system over about $200$ e-folding times. Using nearby
initial conditions, we can construct statistics of the long-term phase-space
structure and compare to rough estimates of resonant widths of the system. We
compared statistics for a wide range of numerical methods, including a
Runge--Kutta method, Wisdom--Holman method, symplectic corrector methods, and a
method by Laskar and Robutel. "Improving" an integrator did not increase the
phase space accuracy, but simply increasing the number of initial conditions
did. In fact, the statistics of a higher order symplectic corrector method were
inconsistent with the other methods in one test.
