  This paper presents new methods and results on almost global synchronization
of coupled Hopf nonlinear oscillators, which are commonly used as the dynamic
model of engineered central pattern generators (CPGs). On balanced graphs, any
positive coupling gain is proven to induce almost global asymptotic
synchronization, and a threshold value for truly global exponential
synchronization is also computed. Furthermore, a hierarchical connection
between coupled Hopf oscillators and Kuramoto oscillators is identified.
Finally, a new result on the synchronization of Kuramoto oscillators with
arbitrary time-varying heterogeneous frequencies and delays is derived.
