Mathematical Model of Microseismic Vibrations Based on Selkov’s Fractional Dynamic System

This chapter proposes a mathematical model built on the basis of Selkov’s nonlinear fractional dynamic system to describe the self-oscillatory modes of microseisms. Selkov’s fractional dynamical system is a system of two nonlinear ordinary differential equations with derivatives of fractional orders, understood in the Gerasimov-Caputo sense, which allows the use of local initial conditions (Cauchy problemCauchy problem). The Cauchy problem describes the interaction of small and large cracks taking into account heredity. Small cracks are not detected by seismic equipment, but an increase in their concentration can lead to the formation of larger cracks, which can already generate microseisms and be recorded. At the same time, large cracks, releasing their energy into the surrounding space, can transform into smaller cracks. As a result, we obtain self-oscillations in the concentration of small and large cracks. The process of transition from one type of crack to another can occur quite slowly due to the effects of heredity. These effects can be described quite well using equations with fractional derivatives. In this paper, the Selkov fractional dynamical systemSelkov fractional dynamical system is studied using the Adams-Bashforth-Moulton numerical algorithmAdams-Bashforth-Moulton numerical algorithm. Oscillograms and phase trajectories were constructed, chaotic and regular modes were studied, and an interpretation of the simulation results was given. Chaotic and regular modes and equilibrium points of the system are studied.