On the initial function space of time-delayed systems: A time-delayed feedback control perspective

We address the problem as if the initial function space of a time-delayed system can be finite-dimensional. We argue that the dimension of the initial function space of a time-delayed feedback control system is just identical to that of the corresponding plant which is to be controlled, and so is finite-dimensional in many cases. We prove this property by constructing an injection map between the two spaces. We anticipate that this may also be applied to some other systems, such as ecological processes involving time delays without control. We believe that the significance of this discovery not only makes available the physical interpretation and intuitional explanation of the basins of attraction associated with time-delayed feedback control, but also suggests the principle as how to evaluate the robustness of the time-delayed feedback control schemes to external disturbances. We finally visualize and illustrate such basins of attraction via numerical simulation for the classical Duffing system by equipping it with the velocity time-delayed feedback control.