Chaotic modes in the low-mode model $\alpha\Omega$-dynamo with hereditary $\alpha$-quenching by the field energy

This article considers the conditions under which it is possible to simulate the chaotic regime of the magnetic field in a large-scale model $\alpha\Omega$-dynamo in a low-mode approximation. The intensity of the $\alpha$– and $\Omega$-generators is regulated by the Lorentz force. The quenching of the $\alpha$-effect is determined by the action of the Lorentz force through a process with hereditarity properties (finite «memory»). The nature of the impact of the process is determined by an alternating kernel with variable damping frequency and damping coefficient. The effect of large-scale and turbulent generators on the magnetohydrodynamic system is embedded in the control parameters — the Reynolds number and the measure of the $\alpha$-effect, respectively. Within the framework of this work, the solutions of the magnetohydrodynamic system are investigated for Lyapunov stability in the vicinity of the rest point, depending on the set values of the input parameters. Based on the results of the numerical experiment, the limitations of the stability characteristic and parameters of the system are determined, under which it is possible to simulate the chaotic regime of the magnetic field.