  The interplay between time scales and structural properties of complex
networks of nonlinear oscillators can generate many interesting phenomena, like
amplitude death, cluster synchronization, frequency synchronization etc. We
study the emergence of such phenomena and their transitions by considering a
complex network of dynamical systems in which a fraction of systems evolves on
a slower time scale on the network. We report the transition to amplitude death
for the whole network and the scaling near the transitions as the connectivity
pattern changes. We also discuss the suppression and recovery of oscillations
and the cross over behavior as the number of slow systems increases. By
considering a scale free network of systems with multiple time scales, we study
the role of heterogeneity in link structure on dynamical properties and the
consequent critical behaviors. In this case with hubs made slow, our main
results are the escape time statistics for loss of complete synchrony as the
slowness spreads on the network and the self-organization of the whole network
to a new frequency synchronized state. Our results have potential applications
in biological, physical, and engineering networks consisting of heterogeneous
oscillators.
