  The symmetric multistep methods developed by Quinlan and Tremaine (1990) are
shown to suffer from resonances and instabilities at special stepsizes when
used to integrate nonlinear equations. This property of symmetric multistep
methods was missed in previous studies that considered only the linear
stability of the methods. The resonances and instabilities are worse for
high-order methods than for low-order methods, and the number of bad stepsizes
increases with the number frequencies present in the solution. Symmetric
methods are still recommended for some problems, including long-term
integrations of planetary orbits, but the high-order methods must be used with
caution.
