The spatial group and cyclic oscillations caused by the power correlation between the moving direction and the phase of a moving oscillator.

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]