Modelling the magnetic field generation modes in the low-mode model of the $\alpha\Omega$-dynamo with varying intensity of the $\alpha$-effect

In the framework of the $\alpha\Omega$-dynamo model a magnetohydrodynamic system (MHD-system) with an included additive correction of the $\alpha$-effect intensity in the form of a function $Z(t)$ is considered. The variation of the $\alpha$-effect intensity with time is determined by the exponential kernel $J(t)$ of the function $Z(t)$. A preliminary analysis of the changes influence in the values of the mass density of external forces and the $\alpha$-effect intensity on the values of the magnetic field and the velocity field is carried out. To simplify the numerical model, the MHD-system was rescaled and the time of the magnetic field dissipation, which is of the order of $10^4$ years, was taken as a unit of time. The time unit choice is determined by the need to study the magnetic field modes on a large time scales. In order to reduce the number of variable parameters, the MHD-system was made dimensionless so that the control parameters are the Reynolds number, which carries information about the large-scale generator, and the amplitude of the $\alpha$-effect, which characterizes the turbulent generator. The results of numerical simulation of the modes of magnetic field generation are displayed on the phase plane of the control parameters. The question of the dynamics of the change in the pattern on the phase plane depending on the waiting time determined by the scale factor of the exponential kernel $J(t)$ of the function $Z(t)$ is investigated.