Models of active Brownian motion in two-dimensional (2D) systems developed earlier are investigated with respect to the influence of linear attracting forces and external noise. Our consideration is restricted to the case that the driving is rather weak and that the forces show only weak deviations from radial symmetry. In this case an analytical study of the bifurcations of the system is possible. We show that in the presence of external linear forces with only small deviations from radial symmetry, the system develops rotational excitations with left-right symmetry, corresponding to limit cycles in the 4D phase space, the corresponding distribution has the form of a hoop or a tire in the 4D space. In the last part we apply the theory to swarms of Brownian particles that are held together by weak and attracting forces, which lead to cluster formation. Since near the center the potential is at least approximately parabolic and near to the radial symmetry, the swarm develops rotational modes of motion with left-right symmetry.

• Received 24 July 2001

DOI:https://doi.org/10.1103/PhysRevE.65.061106