Vibrational Stabilization of Cluster Synchronization in Oscillator Networks

Cluster synchronization is of great importance for the normal functioning of numerous technological and natural systems. Deviations from normal cluster synchronization patterns are closely associated with various malfunctions, such as neurological disorders in the brain. Therefore, it is crucial to restore normal system functions by stabilizing the appropriate cluster synchronization patterns. Most existing studies focus on designing controllers based on state measurements to achieve system stabilization. However, in many real-world scenarios, measuring system states in real time, such as neuronal activity in the brain, poses significant challenges, rendering the stabilization of such systems difficult. To overcome this challenge, in this paper, we employ an open-loop control strategy, vibrational control, which does not require any state measurements. We establish some sufficient conditions under which vibrational inputs stabilize cluster synchronization. Further, we provide a tractable approach to design vibrational control. Finally, numerical experiments are conducted to demonstrate our theoretical findings.