  We consider a quantum system dynamics caused by successive selective and
non-selective measurements of the probe coupled to the system. For the finite
measurement rate $\tau^{-1}$ and the system-probe interaction strength $\gamma$
we derive analytical evolution equations in the stroboscopic limit $\tau
\rightarrow 0$ and $\gamma^2 \tau = {\rm const}$, which can be considered as a
deviation from the Zeno subspace dynamics on a longer timescale $T \sim
(\gamma^2 \tau)^{-1} \gg \gamma^{-1}$. Non-linear quantum dynamics is analyzed
for selective stroboscopic projective measurements of an arbitrary rank.
Non-selective measurements are shown to induce the semigroup dynamics of the
system-probe aggregate. Both non-linear and decoherent effects become
significant at the timescale $T \sim (\gamma^2 \tau)^{-1}$, which is
illustrated by a number of examples.
