  In this paper, we investigate the development of generalized synchronization
(GS) on typical complex networks, such as scale-free networks, small-world
networks, random networks and modular networks. By adopting the
auxiliary-system approach to networks, we show that GS can take place in
oscillator networks with both heterogeneous and homogeneous degree
distribution, regardless of whether the coupled chaotic oscillators are
identical or nonidentical. For coupled identical oscillators on networks, we
find that there exists a general bifurcation path from initial
non-synchronization to final global complete synchronization (CS) via GS as the
coupling strength is increased. For coupled nonidentical oscillators on
networks, we further reveal how network topology competes with the local
dynamics to dominate the development of GS on networks. Especially, we analyze
how different coupling strategies affect the development of GS on complex
networks. Our findings provide a further understanding for the occurrence and
development of collective behavior in complex networks.
