Swarmalators with Discrete-Time Coupling: Which Step Size to Choose?

Implementing swarmalator behavior on mobile robots requires discretizing calculations and interactions. This is accomplished through numerical methods for differential equations, which involve selecting a step size for the density of supporting points. Large steps prevent convergence; small ones increase computational effort and interaction rate. We discuss how to choose an appropriate step size by deriving bounds for divergence, oscillatory behavior, and monotonic convergence, particularly for circular patterns. Our main result is a bound that ensures non-oscillatory convergence, enabling researchers to perform real-world system parameterization.