  Low Mass X-Ray Binaries (LMXBs) are systems in which a compact object
accretes from a binary companion star via an accretion disk. The X-ray
properties of LMXBs show strong variability over timescales ranging from
milliseconds to decades, much of which is tied to the extreme environment of
the inner accretion disk, hence an understanding of this behaviour is key to
understanding how matter behaves in such an environment. GRS 1915+105 and MXB
1730-335 are two LMXBs which show particularly unusual variability. GRS
1915+105 shows a large number of distinct classes of second-to-minute scale
variability, consisting of repeated patterns of dips and flares. MXB 1730 shows
Type II X-ray Bursts; minute-scale increases in X-ray intensity with a sudden
onset and a slow decay. More recently two new objects, IGR J17091-3624 and GRO
J1744-28 have been shown to display similar behaviours. In this thesis I
present a new framework with which to classify variability in IGR J17091. I
perform a comparison study between this source and GRS 1915. In GRS 1915, hard
X-rays lag soft X-rays in all variability classes; in IGR J17091, I find that
the sign of this lag varies between variability classes. Additionally, while
GRS 1915+105 accretes at close to its Eddington Limit, I find that IGR
J17091-3624 accretes at only ~5-33% of its Eddington Limit. I also perform a
study of variability in GRO J1744 and find that it is more complex than in MXB
1730, consisting of at least 4 separate phenomena which may have separate
physical origins. One of these phenomena, `Structured Bursting', consists of
patterns of flares and dips similar to those seen in GRS 1915 and IGR J17091. I
compare these types of variability and discuss the possibility of a physical
link. I also present the alternative hypothesis that Structured Bursting is
caused my 'hiccup accretion' similar to that seen in systems approaching the
propeller regime.
