Stochastic two-mode hereditary model of a cosmic dynamo

The paper is devoted to a class of stochastic two-mode hereditary models of the cosmic dynamo. The models include two magnetic field generators — large-scale and turbulent ($\alpha$-effect). The influence of the magnetic field on the motion of the medium is presented through the suppression of the $\alpha$-effect by a functional of the field components, which introduces memory (hereditary) into the model. The model describes the dynamics of only large-scale components, but takes into account the possible impact of smallscale modes using a stochastic term. This term models the influence of possible spontaneous synchronization of small-scale modes. The paper also presents a numerical scheme for solving the integro-differential equations of the model. The numerical scheme consists of two parts, for the differential part the Adams «predictor-corrector» method of the fourth order is used, and for the integral part the Simpson method.The main result of the work is a generalized model of a dynamo system, with an additive addition of a random correction to the $\alpha$-generator. Taking into account such a correction significantly diversifies the dynamic modes in the model.