  Using the recently published GJF-2GJ Langevin thermostat, which can produce
time-step-independent statistical measures even for large time steps, we
analyze and discuss the causes for abrupt deviations in statistical data as the
time step is increased for some simulations of nonlinear oscillators.
Exemplified by the pendulum, we identify a couple of discrete-time dynamical
modes in the purely damped pendulum equation as the cause of the observed
discrepancies in statistics. The existence, stability and kinetics of the modes
are consistent with the acquired velocity distribution functions from Langevin
simulations, and we conclude that the simulation deviations from physical
expectations are not due to normal, systematic algorithmic time-step errors,
but instead due to the inherent properties of discrete time in nonlinear
dynamics.
