Stochastically-induced dynamics of earthquakes

Abstract

Motivated by an important geophysical application, we analyze the nonlinear dynamics of the number of earthquakes per unit time in a given Earth’s surface area. At first, we consider a dynamical model of earthquakes describing their rhythmic behaviour with time delays. This model comprises different earthquake scenarios divided into three types (A, B, and C) accordingly to various system dynamics. We show that the deterministic system contains stable equilibria and a limit cycle whose size drastically depends on the production rate $\alpha$ of earthquakes and their time-delay effect. As this takes place, the frequency of earthquakes possesses an oscillatory behaviour dependent on $\alpha$. To study the role of $\alpha$ in more detail, we have introduced a white Gaussian noise in the governing equation. First of all, we have shown that the dynamical system is stochastically excitable, i.e. it excites larger-amplitude noise-induced fluctuations in the frequency of earthquakes. In addition, these large-amplitude stochastic fluctuations can alternate with small-amplitude fluctuations over time. In other words, the frequency of earthquakes can change its amplitude in an irregular manner under the influence of white noise. Another important effect is how close the current value of $\alpha$ is to its bifurcation point. The closer this value is, the less noise generates large-amplitude fluctuations in the earthquake frequency.