  We introduce an analytical model based on birth-death clustering processes to
help understanding the empirical log-periodic corrections to power-law scaling
and the finite-time singularity as reported in several domains including
rupture, earthquakes, world population and financial systems. In our
stochastics theory log-periodicities are a consequence of transient clusters
induced by an entropy-like term that may reflect the amount of cooperative
information carried by the state of a large system of different species. The
clustering completion rates for the system are assumed to be given by a simple
linear death process. The singularity at t_{o} is derived in terms of
birth-death clustering coefficients.
