  We study a Kuramoto-like model of coupled identical phase oscillators on a
network, where attractive and repulsive couplings are balanced dynamically due
to nonlinearity in interaction. Under a week force, an oscillator tends to
follow the phase of its neighbors, but if an oscillator is compelled to follow
its peers by a sufficient large number of cohesive neighbors, then it actually
starts to act in the opposite manner, i.e. in anti-phase with the majority.
Analytic results yield that if the repulsion parameter is small enough in
comparison with the degree of the maximum hub, then the full synchronization
state is locally stable. Numerical experiments are performed to explore the
model beyond this threshold, where the overall cohesion is lost. We report in
detail partially synchronous dynamical regimes, like stationary phase-locking,
multistability, periodic and chaotic states. Via statistical analysis of
different network organizations like tree, scale-free, and random ones, and
different network sizes, we found a measure allowing one to predict relative
abundance of partially synchronous stationary states in comparison to
time-dependent ones.
