A Second-Order Particle Swarm Model on a Sphere and Emergent Dynamics

We introduce a new second-order particle swarm model on a sphere and study its emergent behaviors. The proposed particle swarm model can be naturally derived from the first-order swarm model on a sphere by adding inertial effect, and as special cases, it can be reduced to other collective particle models, such as the Kuramoto model with inertia and the inertial spin model. In this paper, we provide several sufficient frameworks for the emergent dynamics of the proposed second-order particle swarm model in attractive and repulsive coupling regimes in terms of the system parameters and the initial data. We also perform several numerical simulations to see the effect of couplings and compare them with our theoretical results.