  Synchronization on the sphere is important to certain control applications in
swarm robotics. Of recent interest is the Lohe model, which generalizes the
Kuramoto model from the circle to the sphere. The Lohe model is mainly studied
in mathematical physics as a toy model of quantum synchronization. The model
makes few assumptions, wherefore it is well-suited to represent a swarm.
Previous work on this model has focused on the cases of complete and acyclic
networks or the homogeneous case where all oscillator frequencies are equal.
This paper concerns the case of heterogeneous oscillators connected by a
non-trivial network. We show that any undesired equilibrium is exponentially
unstable if the frequencies satisfy a given bound. This property can also be
interpreted as a robustness result for small model perturbations of the
homogeneous case with zero frequencies. As such, the Lohe model is a good
choice for control applications in swarm robotics.
