7. Emergence of Oscillations in Networks of Time-Delay Coupled Inert Systems

We discuss the emergence of oscillations in networks of single-input–single-output systems that interact via linear time-delay coupling functions. Although the systems itself are inert, that is, their solutions converge to a globally stable equilibrium, in the presence of coupling, the network of systems exhibits ongoing oscillatory activity. We address the problem of emergence of oscillations by deriving conditions for; 1. 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 time-delay coupled inert systems have a nontrivial oscillatory solution. In addition, we show that a necessary condition for the emergence of oscillations in such networks is that the considered systems are at least of second order.