  In this paper, by extending the concept of Kuramoto oscillator to the
left-invariant flow on general Lie group, we investigate the generalized phase
synchronization on networks. The analyses and simulations of some typical
dynamical systems on Watts-Strogatz networks are given, including the
$n$-dimensional torus, the identity component of 3-dimensional general linear
group, the special unitary group, and the special orthogonal group. In all
cases, the greater disorder of networks will predict better synchronizability,
and the small-world effect ensures the global synchronization for sufficiently
large coupling strength. The collective synchronized behaviors of many
dynamical systems, such as the integrable systems, the two-state quantum
systems and the top systems, can be described by the present phase
synchronization frame. In addition, it is intuitive that the low-dimensional
systems are more easily to synchronize, however, to our surprise, we found that
the high-dimensional systems display obviously synchronized behaviors in
regular networks, while these phenomena can not be observed in low-dimensional
systems.
