Blümel claims that, for immediately below the cut-off for
region C, there exists a nonheating quasiperiodic region of phase space. That is, the root-mean-square displacements from the trap center which are immediately on the ``dense'' side of the heating regime constitute non-heating configurations. This assertion is important because it implies that there is a nonheating, neutral ``buffer'' zone between the crystalline phase (discussed below) and the chaotic phase.[4] Blümel states that the range of
values which form this quasiperiodic regime decreases as Mathieu parameter **q** increases for a given **a**. However, although the width of the non-heating quasiperiodic regime is compressed as the configuration approaches
the Mathieu instability, Blümel conjectures that in stability region A
``there is always a small phase-space region around the crystalline solution in which no heating occurs.''[4] This statement contradicts conclusions drawn by Hoffnagle, et al.: Hoffnagle's research group claimed the existence of reproducible order chaos transitions (displaying hysteresis) as a function of the Mathieu parameters **a**, **q**. The research group claimed, for **a=0**, a well-marked order chaos transition at
and a chaos order transition at . Hoffnagle's work contradicts Blümel's theory on two points: (* a*) Hoffnagle, et al., claim order chaos transitions as a
function of the Mathieu parameters, whereas Blümel's criterion for predicting ordered and chaotic states is the amount of energy in the system (which can be
gauged by ); (* b*) Hoffnagle claims the existence of an order chaos transition at **a=0,** , which is well below the Mathieu instability of , while Blümel asserts the stability of few-ion crystals in Mathieu regime A--- even in the absence of damping--- right up to the Mathieu instability. The stability of crystals will be further investigated after a discussion of the crystalline phase.

The configuration of file ` cloud5A.trp` depicts **5** Mg ions in this non-heating ``quasiperiodic'' regime. The term ``quasiperiodic'' may be somewhat of a misnomer--- in this closely-coupled cloud phase,
non-linear effects are strong. As shown in Fig. , Fourier transforms of position for the non-heating cloud do not yield the sharp, noiseless peaks of truly quasiperiodic behavior, which is composed of discrete frequencies related by an irrational ratio (Figs. , , ). Nevertheless, Fig. * is* dominated by certain frequency ranges, so one might
call this behavior ``pseudoperiodic.''

Fri May 12 10:36:01 EDT 1995