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In this paper, we introduce a programmable particle swarm evolving on a sphere called swarm on sphere system. This model is a polynomial dynamical system obtained from a consensus algorithm that exhibits a rich set of synchronous behaviors with close connections to the Kuramoto model of coupled oscillators, coalition formation in social networks, small-words, integer programming problems such as max-cut, and networks of self-synchronous oscillators with applications to synthetic biology. We prove the generalized Kuramoto model can be obtained as a special case of this model in dimension two. Moreover, we provide formal stability analysis of aligned, bipolar, and dispersed synchronous modes of the system. As a byproduct of this stability analysis, we obtain simple algorithms for programming the weights of the swarm enabling it to exhibit various desired patterns of synchrony. Simulation results are provided that demonstrate 3-D in-phase synchrony, coalition formation for interacting agents with mixed-sign couplings, and dispersal behavior with spatial order
Reza Olfati‐Saber (Sun,) studied this question.