We consider the synchronization problem of dynamical networks with delayed interactions. More specifically, we focus on the stabilization of synchronous equilibria in regular networks where the degrees of all nodes are equal. By studying such control near a Hopf bifurcation, we obtain necessary and sufficient conditions for stabilization. It is shown that the stabilization domains in the parameter space reappear periodically with time-delay. We find that the frequency of reappearance of the control domains is linearly proportional to the number of cycle multipartitions of the network. IntroductionDynamical networks with complex topology are a very powerful approach for the study of large complex systems in various application areas ranging from neuroscience, engineering, to sociology, economics or Earth sciences. From this perspective, real-world systems are modeled as networks of interacting nodes, where the nodes have their own dynamics and influence each other's behavior in complex ways [1][2][3][4][5]. Research in this area combines various application fields with theoretical approaches from dynamical system theory, statistical physics, time series analysis, or graph theory.The networks can range from a few to hundreds of billions of nodes, such as large communication networks or the brain. Connectivity in networks can vary widely, ranging from "all-to-all" to hierarchical or even sparse.When modeling realistic situations, time-delays must be often taken into account for the interactions in dynamical networks, as is the case in neural systems [6-9], laser networks [10-12], machine learning [13], traffic dynamics [14], and other applications. Particularly challenging are the situations in which the interplay between time-delays and network symmetries [15][16][17][18] or large delays [19-22] lead to new dynamical phenomena. They include synchronization [21,[23][24][25][26][27][28][29][30], explosive synchronization [31,32], various resonances [33], chimera states [24,34] and other patterns [35][36][37][38][39].The interplay between network topology and time-delayed interactions are also explored in [40,41].Many natural systems possess unstable states, and the stabilization of them might be of interest for applications. For example, stabilization of unstable equilibrium solution has application in semiconductor lasers [42], in nanoeletronics [43], in medicine [44], and others. An efficient strategy to control unstable periodic orbits, which is also applied to stabilization of equilibrium, is introducing a self-feedback delayed term. This approach is non-invasive, since the feedback term does not alter the original solutions. In particular, for stabilization of periodic orbits, the usual approach is to consider the delay equal or close to the period of the orbit, known as Pyragas control [45]. For stabilization of equilibrium, the aim can also be achieved with a suitable choice of delay [46,47].Here we consider a network model in which the nodes are interacting with a common time-delay.Hence, the delay is not an...

We consider the synchronization problem of dynamical networks with delayed interactions. More specifically, we focus on the stabilization of synchronous equilibria in regular networks where the degrees of all nodes are equal. By studying such control near a Hopf bifurcation, we obtain necessary and sufficient conditions for stabilization. It is shown that the stabilization domains in the parameter space reappear periodically with time-delay. We find that the frequency of reappearance of the control domains is linearly proportional to the number of cycle multipartitions of the network.