Can playing Spirograph lead to an ordered structure in self-propelled particles?

Understanding the local dynamics of microorganisms infecting a cell could help us develop efficient strategies to counter their aggregation. In the present study we have introduced a simple model of self-propelled particles (SPPs) with constant linear velocity, both in 2 and 3 dimensions, which captures the essential features of a microorganism's aggregation as well the dynamics around an attractive point (AP). The static behavior shows the presence of an icosahedral structure for a finite number of SPPs, and a hexagonal closed packed structure for an infinite number of SPPs, which was confirmed using Steinhardt bond order parameters for a 3-dimensional model. For a single SPP the dynamic behaviour involves the formation of orbits around the AP, which can be categorised into three dynamical regions based on the strength of coupling between the AP and SPP. For weak coupling we observe a rosette-like trajectory reminiscent of the pattern formed by the Spirograph toy. For intermediate coupling, circular trajectories were observed, and for very strong coupling the SPP was static and was always aligned with the AP. The radial distance from the AP to SPP was determined by the angular velocities of the SPP for the rosette-like region whereas for the circular and static regions, it was determined by the coupling constant. Even for a finite number of SPPs we observed the same behavior as long as the SPPs could rotate around the AP without colliding with each other.