Synchronous oscillations in networks of time-delay coupled inert systems

We discuss the emergence of oscillations in networks of inert single-input-single-output systems which interact via linear, time-delay coupling functions. We present conditions for 1. the solutions of the time-delay coupled systems to be bounded, 2. the network equilibrium to be unique, and 3. the network equilibrium to be unstable. If these conditions are all satisfied the network of time-delay coupled inert systems has a non-trivial oscillatory solution. In addition, we show that a necessary and sufficient condition for the emerge of oscillations in such networks is that the considered systems are at least second order. Moreover, using recent results on the existence of partial synchronization manifolds, we identify the patterns of oscillation that may emerge.