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We present a detailed X-ray timing analysis of the highly variable NLS1galaxy, IRAS 13224-3809. The source was recently monitored for 1.5 Ms withXMM-Newton which, combined with 500 ks archival data, makes this the beststudied NLS1 galaxy in X-rays to date. We apply standard time- andFourier-domain in order to understand the underlying variability process. Thesource flux is not distributed lognormally, as would be expected for accretingsources. The first non-linear rms-flux relation for any accreting source in anywaveband is found, with $\mathrm{rms} \propto \mathrm{flux}^{2/3}$. The lightcurves exhibit significant strong non-stationarity, in addition to that causedby the rms-flux relation, and are fractionally more variable at lower sourceflux. The power spectrum is estimated down to $\sim 10^{-7}$ Hz and consists ofmultiple peaked components: a low-frequency break at $\sim 10^{-5}$ Hz, withslope $\alpha < 1$ down to low frequencies; an additional component breaking at$\sim 10^{-3}$ Hz. Using the high-frequency break we estimate the black holemass $M_\mathrm{BH} = [0.5-2] \times 10^{6} M_{\odot}$, and mass accretion ratein Eddington units, $\dot m_{\rm Edd} \gtrsim 1$. The non-stationarity ismanifest in the PSD with the normalisation of the peaked components increasingwith decreasing source flux, as well as the low-frequency peak moving to higherfrequencies. We also detect a narrow coherent feature in the soft band PSD at$0.7$ mHz, modelled with a Lorentzian the feature has $Q \sim 8$ and an$\mathrm{rms} \sim 3$ %. We discuss the implication of these results foraccretion of matter onto black holes.

The remarkable X-ray variability of IRAS 13224-3809 I: The variability process