  Sufficient conditions for synchronization of coupled Lienard-type oscillators
are investigated via averaging technique. Coupling considered here is pairwise,
unidirectional, and described by a nonlinear function (whose graph resides in
the first and third quadrants) of some projection of the relative distance
(between the states of the pair being coupled) vector. Under the assumption
that the interconnection topology defines a connected graph, it is shown that
the solutions of oscillators can be made converge arbitrarily close to each
other, while let initially be arbitrarily far apart, provided that the
frequency of oscillations is large enough and the initial phases of oscillators
all lie in an open semicircle. It is also shown that (almost) synchronized
oscillations always take place at some fixed magnitude independent of the
initial conditions. Similar results are generated for nonlinearly-coupled
harmonic oscillators.
