Our 2026 paper introduced a mathematical framework to identify when and how disorder can be exploited to promote, rather than inhibit, stability in complex systems. The paper discussed many applications ranging from oscillator synchronization (power grids and coupled lasers), to collective motion (drone swarmming and animal flocking) and pattern formation (metamaterials and neuronal networks). In this interactive explorable, we bring synchronization and swarming together into a single model—known as swarmalators —as a demonstration of disorder-promoted stability!
Swarmalators are interacting agents that can both swarm through space and synchronize their internal states. Swarming is represented by the consensus among agents' positions and velocities, while synchronization is represented by the alignment of their arrows (their internal phases). The agents continually adjust both their positions and phases according to the underlying network of interactions. Depending on the balance among attraction, repulsion, and phase-coupling forces, swarmalators may self-organize into coherent patterns, or remain in irregular, disorganized motion.
For both 2D and 1D swarmalators, this explorable demonstrates that disorder can promote stable self-organized behavior. The agents colors encode their parameters; thus, the more colorful, the more heterogeneous they are. When all agents are identical (no disorder), the swarmalators settle into irregular, unstable motion. Introducing an intermediate amount of disorder stabilizes their dynamics, driving them toward a symmetric, self-organized pattern. Too much disorder, however, destroys this organization, and the population separates into two groups, one synchronized and the other desynchronized, forming a chimera state. Play with the disorder knob in the control panel to see this!
The result is quite counterintuitive and interesting: with symmetric (identical) parameters, the system converges toward an asymmetric (disorganized) state; but with asymmetric (disordered) parameters, it converges to a symmetric (organized) state. Hence, disorder promotes stability.