Synchronization in networks of genetic oscillators with delayed coupling

This paper studies the global synchronization problem for gene networks of coupled oscillators with a coupling delay. Several synchronization conditions are obtained for coupled genetic oscillators by means of Lyapunov functional theory and matrix inequality approach. More specifically, two conditions are first presented in terms of matrix inequalities, under which the synchronization can be achieved irrespective of how large the coupling delay is. Then, a sufficient condition is further proposed to ensure the synchronization of coupled genetic oscillators for a certain range of coupling delays. A numerical example of coupled Goodwin oscillators is given to illustrate the effectiveness of these conditions. Both theoretical and numerical results show that the coupling delay can affect the dynamic behaviors of coupled genetic oscillators, and the synchronized state can be significantly different from that of a single oscillator when a coupling delay is present.