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Our previous work investigated the synchronous dynamics of moving phase oscillators in a two-dimensional plane with periodic boundaries. A mobile phase oscillator can only couple to adjacent oscillators located within its communication radius. In our earlier research, we assumed that the agents' movement direction is identical to the oscillators' phase. However, in our current study, we have made a different assumption: the direction of an agent's movement conforms to a power function of the phase of an oscillator. By configuring various power functions, we can restrict the maximum angle at which the agents can navigate, thereby controlling the agent's range of movement. To explore this new assumption, we performed numerical simulations and analyzed both macroscopic and microscopic characteristics of collective dynamics among locally connected oscillators for various power exponents. Our findings revealed a novel phenomenon of cyclic oscillation during the phase transition to synchronization at the macroscopic level. On a microscopic level, initially formed three clusters gradually merge into a single cluster which eventually achieves full synchronization as coupling strength increases. [ABSTRACT FROM AUTHOR]

Lots of studies on synchronization focus on networks with static topologies, but ignore the synchronization in time-varying networks. We study the synchronization of the system in which agents carrying phase oscillators move in two-dimensional lattice with periodic boundary conditions and only interact with their neighbors. In particular, we assume that the direction of movement of an agent is positively correlated to the phase of the oscillator. By numerical simulations, we find cyclic oscillations in the phase transition to synchronization. Remarkably, the oscillators cluster spatially toward the oscillators with higher natural frequency to form groups. And the groups will collapse from the center, as coupling strength exceeds the critical value. In addition, we investigate the effect of moving speed on synchronization and find that the cyclic oscillation during synchronization will disappear with the increment of moving speed. • The direction of the moving Kuramoto oscillator was correlated with its phase. • The cyclic oscillation of order parameter will emerge for a larger coupling strength. • The oscillators' frequency distribution will affect the formation of the final groups. • The cyclic oscillations will disappear as the speed of movement increases. [ABSTRACT FROM AUTHOR]