Attractive Domain of Nonlinear Systems with Time-delayed Feedback Control and Time-delay Effects

The dynamics of system with time delay resides in an infinite dimensional state space. It is difficult to determine the domain of attraction of the steady state response in the infinite dimensional state space. A definition of the domain of attraction in the physical space is introduced, which is intuitive, and furthermore, it is numerically computable under the hypothesis that the time delay is short. The effect of time delay on the domain of attraction of the popular Duffing system is considered. It is found that the time delay has significant influence on the structure of domain of attraction. The width of attractive domain can be enlarged with a small time delay, however, the domain of attraction shrinks when the time delay increases and approach to the stability boundary.