Computationally implementable sufficient conditions for the synchronisation of coupled dynamical systems with time delays in the coupling

Understanding the mechanisms behind synchronisation of coupled systems is important in biology, chemistry, physics and engineering. For example, research on the synchronisation of coupled circadian clocks is of much importance to better understand many different disease states. Or, different applications in engineering require certificates for synchronised behaviour between the different coupled subsystems. Such systems could correspond to tethered space vehicles or to vehicles in formation flight. In this paper, we provide sufficient conditions for the synchronisation of coupled identical nonlinear dynamical systems with time delays in the coupling, which makes the modelling more realistic. Importantly, the conditions can be efficiently checked using semi-definite programming techniques, which we illustrate through applications.