Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay

When a transmission delay occurs in the interconnection of linearly coupled systems described by ordinary differential equations (LCODEs), both synchronization and the final synchronized state will vary. In this paper, mathematical analysis is presented on the synchronization phenomena of LCODEs with a single coupling delay. Criteria are derived for both local and global synchronization. It is known that in addition to the dynamical behaviors of the underlying uncoupled system and the coupling configuration, the coupling strength and the coupling delay also play key roles on the stability of synchronization. Both theoretical and numerical analysis indicate that under some conditions, if the coupling strength is large enough, the coupled system can be completely synchronized for any coupling delay. On the other hand, in some cases, the coupled system can be synchronized if the coupling delay is small enough.