Windows of opportunity in blinking complex networks: Synchronizability under intermediate switching frequencies

Blinking complex networks can enable synchronization at a lower cost, i.e., with lower coupling strength. Most current work focuses on fast or slow switching regimes. However, there are conditions under which neither of them can help. In this study, we derive the conditions under which neither static nor fast- or slow-blinking complete networks synchronize. By selecting network parameters under these conditions, we show that windows of opportunity can emerge in which blinking complete networks can synchronize while blinking at intermediate frequencies. This phenomenon is observed by calculating the synchronization error and synchronization probability as the blinking period is increased from low to high. Intermediate-frequency synchronization is established numerically, while the theoretical derivations serve solely to restrict the parameter domain. To improve control over Laplacian eigenvalues, we report results for complete networks with 2 to 10 nodes that blink with both similar and different on and off times. We show the emergence of these windows in complete networks of chaotic Hindmarsh-Rose neurons with 2-1 electrical coupling, Lorenz oscillators with 2-1 electrical coupling, and Rössler oscillators with 1-1 electrical coupling. The Chen oscillator with 3-3 electrical coupling is presented as a boundary case, showing how subtle changes in the Master Stability Function (MSF) curvature lead to fragile, discontinuous synchronization rather than windows of opportunity. All these systems and couplings have a type 2 MSF with negative curvature at the first zero-crossing, a previously suggested indicator of windows of opportunity. Time-series and phase-portrait analyses further validate the collective behavior of blinking complete networks with slow, intermediate, and fast blinking frequencies.