Emergent behaviors of a holonomic particle system on a sphere

We study sufficient conditions for the asymptotic emergence of synchronous behaviors in a holonomic particle system on a sphere, which was recently introduced by Lohe ["Non-Abelian Kuramoto model and synchronization," J. Phys. A: Math. Theor. 42, 395101-395126 (2009)]. These conditions depend only on the coupling strength and initial position diameter. For identical particles, we show that the position diameter approaches zero asymptotically under these sufficient conditions, i.e., all particles approach to the same position. 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. We also provide several numerical results to confirm our analytical findings.