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We present our analysis of the X-ray variability of two ultraluminous X-raysources (ULXs) based on multiple XMM--Newton observations. We show the linearrms-flux relation is present in eight observations of NGC5408 X-1 and also inthree observations of NGC6946 X-1, but data from other ULXs are generally notsufficient to constrain any rms-flux relation. The presence of this relationwas previously reported in only two observations of NGC 5408X-1; our resultsshow this is a persistent property of the variability of NGC5408 X-1 andextends to at least one other variable ULX. We speculate this is a ubiquitousproperty of ULX variability, as it is for X-ray variability in other luminousaccreting sources. We also recover the time delay between hard and soft bandsin NGC5408 X-1, with the soft band (<1 keV) delayed with respect to the hardband (>1 keV) by up to ~10 s (~0.2 rad) at frequencies above ~few mHz. For thefirst time, we extend the lag analysis to lower frequencies and find someevidence for a reversal of the lag, a hard lag of ~1 ks at frequencies of ~0.1mHz. Our energy-resolved analysis shows the time delays are energy dependent.We argue that the lag is unlikely to be a result of reflection from anaccretion disc (`reverberation') based on the lack of reflection features inthe spectra, and the large size of the reflector inferred from the magnitude ofthe lag. We also argue that associating the soft lag with a quasi-periodicoscillation (QPO) in these ULXs - and drawing an analogy between soft lags inULXs and soft lags seen in some low-frequency QPOs of Galactic X-ray binaries -is premature.

X-ray time lags and nonlinear variability in the ultraluminous X-ray sources NGC 5408 X-1 and NGC 6946 X-1