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.