The first stability tests applied sudden, discontinuous changes of 
to undamped two- and four-ion crystals. In both cases, the crystalline structure was destroyed well before the Mathieu instability. For the two-ion crystal, the alternating voltage was instantaneously changed from 
V (
) to 
V (
); after equilibration, 
was raised to 
V (
), then to
V (
). As shown in Tab. 
, 
decreased as 
was raised, while the amplitude of radial oscillations due to micromotion increased. When the crystal melted at 
, there was a gain in kinetic energy which, averaged over 
s, amounted to a heating rate of 
. The melted two-ion configuration is in file 2sudden.trp.
Table shows the trend of decreasing 
as q was raised discontinuously for a planar four-ion crystal. Like the two-ion crystal, the four-ion crystal melted far before 
reached the Mathieu instability:
after 
s of simulation at 
with 
m, the crystal's kinetic energy and 
increased substantially.
These results demonstrate that crystals in region A are not stable
when sudden changes in Mathieu parameters are applied. Qualitatively, this
``instability'' can be understood by noting that sudden changes in the Mathieu
parameters disrupt the phase and amplitude of the crystal's micromotion oscillations and that quantitative features of the crystal (i.e., 
) are Mathieu parameter-dependent. When a Mathieu parameter undergoes a sudden change, the crystalline configuration is no longer a steady-state (analogous to an eigenstate) solution for the given system. Rather, it may be interpreted as a superposition of solutions, some of which may correspond to heating configurations and enable melting via diffusive energy gain from
the oscillating potential.
Table: Structural response of 2-ion crystal to instantaneous changes in
magnitude of applied oscillating voltage, 
. Note the
increase in micromotion amplitude with increasing magnitude of the
oscillating potential, 
. source file: 2CrA.trp.
Table: Change in coupling parameter (
) of a 4-ion crystal in response to instantaneous changes in magnitude of applied oscillating voltage, 
. source file: 4CrA.trp.