Synchronization and dynamical system and its generalization
Sun-ho Choi (KAIST) /
First, we study sufficient conditions for the asymptotic emergence of synchronous behaviors in a holonomic particle system on a sphere, which was recently introduced by M. Lohe. For identical particles, we show that the position diameter approaches zero asymptotically under these sufficient conditions. For non-identical particles, the particle positions do not shrink to one point, but can be squeezed into some small region whose diameter is inversely proportional to the coupling strength, when the coupling strength is large. Second, we introduce an order parameter measuring the degree of synchronization. Considering the identical oscillators, we show that the order parameter evolves from non-zero values toward the unit value or zero exponentially fast, depending on the nature of the couplings. Finally, we prove that the state to the Lohe system converges to the ground state with zero order parameter for identical oscillators with repulsive couplings and we study a time delayed interaction case and more generalizations.