Synchronization and control in networks with strongly time-delayed couplings - Marcos Daniel Nogueira Maia (2017)
The stability of synchronization and control in networks of dynamical systems with strongly delayed connections is investigated. Strict conditions for both, synchronization of stable periodic and equilibrium solutions , and control of unstable equilibrium are obtained. With a network model including self-feedback delay, the existence of a critical coupling strength kc is demonstrated, which is related to the network structure, isolated vector field and coupling function, such that for large delay and coupling strength k < kc the network undergoes to stable synchronization. Moreover, it is derived that for heterogeneous networks, kc $→$ 0 as the network size grows to infinity, unless the coupling parameter scales with the maximum degree. In contrast, for random networks, the interval of coupling strengths that leads to stable synchronization is the maximum possible when the connectivity threshold is crossed making the network connected. Based on the network structure, the scaling of the coupling parameter, which allows for a synchronization, is derived. And, with a network model consisting of instantaneous self-connections, it is shown that it is possible to stabilize synchronous equilibrium that is unstable in an isolated system. Such a control close to a Hopf bifurcation is studied in details and strict conditions for the stability are obtained. In particular, it is demonstrated that the stabilization domains in parameter space are reappearing periodically and decreasing in size with the increase of time-delays. Also, the frequency of the reappearance of the control domains and the number spectral roots of the adjacency matrix are closely dependent, for instance, the number of cycle multi-partitions of the graph indicates the reappearance frequency of the control domains.