IGOR BELYKH
Department of Mathematics and Statistics, Georgia State University, 30 Pryor Street, Atlanta, GA 30303, USA
MARTIN HASLER
Author for correspondence.
School of Computer and Communication Sciences, Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 14, 1015 Lausanne, Switzerland
VLADIMIR BELYKH
Department of Mathematics, Volga State Academy, 5, Nesterov St., Nizhny Novgorod 603 600, Russia

Received October 18, 2005
Revised March 6, 2006

We review and illustrate our recent results on globally stable synchronization in directed oscillator networks. We consider asymmetrically connected networks with node balance, the property that the sum of the coupling coefficients of all edges directed to a node equals the sum of the coupling coefficients of all the edges directed outward from the node. We show that for such directed but node balanced networks, it is sufficient to symmetrize all connections by replacing a unidirectional coupling with a bidirectional coupling of half the coupling strength. The synchronization condition for the symmetrized network then guarantees synchronization in the original directed network. By considering an example of coupled driven pendula, we show how to prove global stability of synchronization in a concrete unidirectional network. We also discuss the relation between local and global synchronization.

Synchronization; stability; directed networks; node balance