Abstract
In this letter, we find how the frequency of an oscillation determines the exact form of the control for suppressing the oscillation through feedback controls with time delays. These results are based on necessary and sufficient conditions we analytically established for the stability of a dynamical system with feedback control and time delays. We also interpret how these conditions change as the time delay either is equal to zero or becomes larger appropriately. All the analytical and numerical results are illustrated by suppressing the oscillations of the FitzHugh-Nagumo model and by the oscillation death and synchronization phenomena observed in a complex dynamical network with time-delayed couplings. Our findings could be potentially useful for modulating oscillations through proper control devices in various fields.
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