  We present a detailed X-ray timing analysis of the highly variable NLS1
galaxy, IRAS 13224-3809. The source was recently monitored for 1.5 Ms with
XMM-Newton which, combined with 500 ks archival data, makes this the best
studied NLS1 galaxy in X-rays to date. We apply standard time- and
Fourier-domain in order to understand the underlying variability process. The
source flux is not distributed lognormally, as would be expected for accreting
sources. The first non-linear rms-flux relation for any accreting source in any
waveband is found, with $\mathrm{rms} \propto \mathrm{flux}^{2/3}$. The light
curves exhibit significant strong non-stationarity, in addition to that caused
by the rms-flux relation, and are fractionally more variable at lower source
flux. The power spectrum is estimated down to $\sim 10^{-7}$ Hz and consists of
multiple peaked components: a low-frequency break at $\sim 10^{-5}$ Hz, with
slope $\alpha < 1$ down to low frequencies; an additional component breaking at
$\sim 10^{-3}$ Hz. Using the high-frequency break we estimate the black hole
mass $M_\mathrm{BH} = [0.5-2] \times 10^{6} M_{\odot}$, and mass accretion rate
in Eddington units, $\dot m_{\rm Edd} \gtrsim 1$. The non-stationarity is
manifest in the PSD with the normalisation of the peaked components increasing
with decreasing source flux, as well as the low-frequency peak moving to higher
frequencies. 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 for
accretion of matter onto black holes.
