Stochastic theory of log-periodic patterns

We introduce an analytical model based on birth-death clustering processes to help in understanding the empirical log-periodic corrections to power la w scaling and the finite-time singularity as reported in several domains in cluding rupture, earthquakes, world population and financial systems. In ou r stochastic theory log-periodicities are a consequence of transient cluste rs induced by an entropy-like term that may reflect the amount of co-operat ive 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 to is derived in terms o f birth-death clustering coefficients.