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.
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