Our theoretical and numerical analysis have suggested that for low-mass main sequences stars (of the spectral classes from M5 to G0) rotating much faster than the Sun, the generated large-scale magnetic field is caused by the mean-field α2 dynamo, whereby the α2 dynamo is modified by a weak differential rotation. Even for a weak differential rotation, the behaviour of the magnetic activity is changed drastically from aperiodic regime to non-linear oscillations and appearance of a chaotic behaviour with increase of the differential rotation. Periods of the magnetic cycles decrease with increase of the differential rotation, and they vary from tens to thousand years. This long-term behaviour of the magnetic cycles may be related to the characteristic time of the evolution of the magnetic helicity density of the small-scale field. The performed analysis is based on the mean-field simulations (MFS) of the α2 and α2 dynamos and a developed non-linear theory of α2 dynamo. The applied MFS model was calibrated using turbulent parameters typical for the solar convective zone. © 2023 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society.