We employ the swarmalator model to make a group of entities align in a self-organized way along one or more circles.
Written by Christian Bettstetter. Illustration created using ChatGPT.
Many animals organize into well-structured spatial patterns, like flocks of birds and swarms of insects. To understand and analyze such collective motion, scientists have developed mathematical models describing how entities interact, coordinate, and move. One such approach is the swarmalator model, short for “swarming oscillator,” whose unique feature is to combine spatial coordination with the alignment of internal oscillator phases. “Swarmalators can be configured to form ring-shaped patterns, where all entities arrange along a circle and order themselves according to their phase,” explains Marcus Schref, a researcher at the University of Klagenfurt and co-author of a new study on the topic. Together with his advisors, Christian Bettstetter and Udo Schilcher, Schref analyzed some key properties of circular swarmalator patterns, deriving their radii and examining stability. A new insight was that swarmalators can arrange themselves not only on a single circle or inside an annulus, but also on multiple discrete concentric circles. The findings will be published in Physical Review E — a journal for research on nonlinear dynamics and emergent behavior. “Using mathematical analysis and simulation,” Schilcher explains, “we found the conditions for which a single-circle pattern loses its stability and breaks into several concentric circles.”
These results could be essential for building real-world swarmalator systems, helping engineers choose suitable parameters to ensure that the desired patterns indeed emerge and remain stable. A typical application can be found in the domain of autonomous robotics: Consider a group of drones that arranges in a circle around a high-rise building that caught fire, providing a seamless 360-degree view to emergency crews. There’s no pilot guiding them, no central controller issuing directions, and no pre-programmed waypoints or flight paths — just dozens of aerial machines coordinating their movements like schools of fish or herds of cattle. Formations like these have broader potential, from monitoring industrial sites to drone swarms around aircraft.
There is an important difference between the drone swarms we have seen in commercial aerial light shows for some years and those being developed in Klagenfurt. The drones in light shows are not true swarms, as they are usually pre-programmed to follow fixed trajectories. In contrast, systems based on swarmalators or related models are founded on the principles of self-organization and emergence. The formed patterns are not predefined but arise from local interactions between the drones. As a result, the system can adapt in real time to changes like drone failures and changing mission goals.
Self-organizing networked systems were long considered a purely academic research topic. In recent years, however, an increasing number of industries have begun to recognize their potential and are now exploring their application in the real world. The concepts of swarm formation and swarm intelligence are increasingly being adopted in sectors such as production, logistics, and defense.
Publication
Udo Schilcher, Marcus Schref, and Christian Bettstetter. Multicircular Static Patterns in Swarmalator Systems: Radii and Stability. Accepted for Physical Review E. Open access, May 2025. Download author’s preprint.
This research was funded in whole or in part by the Austrian Science Fund (FWF) (grant 10.55776/P30012). Some text passages in this blog article stem from the above-mentioned publication, mainly in modified form. Suggestions for language improvement were generated using ChatGPT.