  We investigate the engineering scenario where the objective is to synchronize
heterogeneous oscillators in a distributed fashion. The internal dynamics of
each oscillator are general enough to capture their time-varying natural
frequency as well as physical couplings and unknown bounded terms. A
communication layer is set in place to allow the oscillators to exchange
synchronizing coupling actions through a tree-like leaderless network. In
particular, we present a class of hybrid coupling rules depending only on local
information to ensure uniform global practical or asymptotic synchronization,
which is impossible to obtain by using the Kuramoto model customarily used in
the literature. We further show that the synchronization set can be made
uniformly globally prescribed finite-time stable by selecting the coupling
function to be discontinuous at the origin. Novel mathematical tools on
non-pathological functions and set-valued Lie derivatives are developed to
carry out the stability analysis. The effectiveness of the approach is
illustrated in simulations where we apply our synchronizing hybrid coupling
rules to models of power grids previously used in the literature.
