Intrinsic Stability, Time Delays and Transformations of Dynamical Networks

Abstract

The dynamics of most real networks are time delayed. These time delays are important as they can often have a destabilizing effect on the network’s dynamics. Here we introduce the notion of intrinsic stability, which is stronger than the standard notion of stability used in the study of network dynamics or more generally in the study of multidimensional systems. The difference between a stable and an intrinsically stable network is that an intrinsically stable network maintains its (intrinsic) stability as nondistributed time-delays are incorporated into or removed from the network. Importantly, determining whether a network is intrinsic stability is fairly easy for concrete systems. Therefore, one can analyze the stability of a time-delayed network by analyzing the stability of the much simpler undelayed version of the network. One can simplify this analysis even further by removing the network’s implicit time-delays, which further reduces the network to a lower-dimensional system which can be used to gain improved stability estimates of the original unreduced network. A network can also be expanded in ways that preserves its intrinsic stability, which suggests how a network with an evolving structure can maintain its stability as it grows. Thus, intrinsic stability is the property that ensures the stability not only of a network but also of its reductions, expansions, and its time-delayed variants. We illustrate these results and techniques on various examples of Cohen–Grossberg neural networks.