On global properties of time-delayed feedback control: weakly nonlinear analysis

We address the problem which type of initial state yields successful stabilisation when time-delayed feedback control is applied. Using well known concepts from bifurcation theory we provide a general mechanism that limits the corresponding basins of attraction. Application of normal form analysis to general delay-differential equations proves the universality of our results. The empirical criteria which determine the prospective size of the basin are easily accessible and a demonstration in terms of a simple numerical simulation is provided.