  Self-propelled point-like particles move along circular trajectories when
their translocation velocity is constant and the angular velocity related to
their orientation vector is also constant. We investigate the collective
behavior of ensembles of such circle swimmers by Brownian dynamics simulations.
If the particles interact via a "velocity-trajectory coordination" rule within
neighboring particles, a self-organized vortex pattern emerges. This vortex
pattern is characterized by its particle-density correlation function $G_\rho$,
the density correlation function $G_c$ of trajectory centers, and an order
parameter $S$ representing the degree of the aggregation of the particles.
Here, we systematically vary the system parameters, such as the particle
density and the interaction range, in order to reveal the transition of the
system from a light-vortex-dominated to heavy-vortex-dominated state, where
vortices contain mainly a single and many self-propelled particles,
respectively. We also study a semi-dilute solution of curved,
sinusoidal-beating flagella, as an example of circling self-propelled particles
with explicit propulsion mechanism and excluded-volume interactions. Our
simulation results are compared with previous experimental results for the
vortices in sea-urchin sperm solutions near a wall. The properties of the
vortices in simulations and experiments are found to agree quantitatively.
