Phase-locking and finite collisions in discrete-time Kuramoto oscillators

Synchronization of coupled oscillators is a fundamental phenomenon in physics, exemplified by the Kuramoto model. In many practical systems (robotic networks, power grids) and numerical simulations, the dynamics are inherently discrete. We prove rigorously that in the discrete-time Kuramoto model, phase locking is achieved if and only if only finitely many oscillator collisions occur. Under a small step-size regime (κh≤1), collision events become the sole factor determining convergence. In particular, for identical-frequency oscillators every trajectory is collision free and thus synchronizes. For heterogeneous frequencies, we derive explicit bounds on initial phase spread and frequency differences that guarantee eventual phase locking. Conversely, large time steps can prevent synchronization: we show even two identical oscillators fail to lock if κh>2. Extensive numerical simulations illustrate both phase-locking and divergent behavior under different conditions. The finite-collisions perspective clarifies synchronization criteria in discrete networks and suggests collision avoidance as a practical diagnostic for robust timing in digital oscillator systems such as swarm robotics or microgrid control systems.