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.

Motivated by important applications in materials science and biology, we consider a non-stationary coagulation equation with in-flux and out-flux of particles (cargo). An exact analytical solution to this integrodifferential equation is found in a parametric form. This solution behaves as a decaying exponent with respect to the particle volume (fluorescence intensity in the case of coagulation of endosomes). We demonstrate that our exact solution substantially depends on coagulation kernels (fusion and fission rates) as well as the in-flux and out-flux processes of cargo.

The article is concerned with the analytical solution to the integro-differential system of balance and kinetic equations that describe the crystal growth phenomenon in a binary system for various nucleation kinetics. The effect of impurity concentration on the evolutionary behavior of crystals is shown. The nonlinear dynamics of a supercooled binary melt is studied with allowance for the withdrawal mechanism of product crystals from a metastable liquid of the crystallizer.