  We present a detailed spectral and temporal study of the intermediate-type
blazar ON 231 during the TeV outburst phase in 2008 June with observations
performed by Swift and XMM-Newton. The X-ray flux of the source, which was
significantly dominated by the soft photons (below $3-4$ keV), varies between
27$\%$ and 38$\%$ on day timescales, while mild variations were observed in the
optical/UV emissions. We found a maximum soft lag of $\sim 1$ hr between the UV
and soft X-ray bands, which can be understood if the magnetic field of the
emitting region is $\sim 5.6~ \delta^{-1/3}$ G. The $0.6-10$ keV spectra can be
well represented by a broken power-law model, which indicates the presence of
both synchrotron and inverse Compton components in the studied X-ray regime.
The synchrotron part of the SEDs constructed with simultaneous optical/UV and
X-ray data follows a log-parabolic shape. A time-resolved spectral analysis
shows that the break energy varies significantly between 2.4 and 7.3 keV with
the changing flux state of the source, and the similar variations of the
spectral slopes of the two components support the SSC scenario. The synchrotron
tail, following a log-parabolic function, shows that the peak frequency
($\nu_{p}$) varies by two orders of magnitude ($\sim 10^{14}-10^{16}$ Hz)
during the event. A significantly positive $E_{p}-\beta$ relation is observed
from both SED and time-resolved spectral analyses. The most feasible scenario
for the observed trend during the flaring event could be associated with a
magnetic-field-driven stochastic process evolving toward an equilibrium energy
level.
