Equation of the Aftershocks and its Phase Portrait

Abstract

Recently, simple differential models of aftershocks have attracted increased attention of researchers at the Institute of Physics of the Earth, Russian Academy of Sciences for their remarkable properties. The shortened Bernoulli equation allows us to introduce the concept of the source deactivation factor and phenomenologically describe the nonstationarity of the geological environment in the earthquake source, "cooling down" after the main shock. The Verhulst logistic equation makes it possible to take into account the transition of the aftershock flow to the background seismicity mode. The Kolmogorov-Petrovsky-Piskunov nonlinear diffusion equation is convenient for modeling the spatio-temporal evolution of aftershocks. This paper focuses on the phase portrait of a dynamical system that simulates the evolution of aftershocks. The phase portrait allows us to understand why the Omori hyperbolic law is most clearly manifested after strong earthquakes. It is noted that the Omori epoch, during which the deactivation coefficient is constant, does not immediately end with a transition to the background mode. In the experience after the end of the Omori epoch, the deactivation coefficient changes over time in a rather complex way, and can even becomes negative. The theory predicts that if negative values of the coefficient are held long enough, then a geotectonic explosion occurs, vaguely resembling the main shock of earthquake. The paper discusses possible directions for the further development of the theory. Keywords: earthquake, aftershock, deactivation factor, Omori epoch, logistic equation, explosive instability, Bath law.