ANALYSIS AND COMPARISON OF MULTIPLE-DELAY SCHEMES FOR CONTROLLING UNSTABLE FIXED-POINTS OF DISCRETE MAPS

We investigate theoretically the stabilization of a fixed point of a d iscrete one-dimensional nonlinear map by applying small perturbations to an accessible system parameter or variable. The size of the perturb ations is determined in real time using feedback schemes incorporating only the dynamical state of the system and its state at previous iter ates without making a comparison to a reference state. In particular, we compare and contrast two algorithms: extended time-delay autosynchr onization, which uses an infinite series of past iterates with weights that decay by a factor of R with each time step, and N-time-delay aut osynchronization, which uses an average of N past iterates with equal weights. The range of feedback parameters that successfully stabilize the fixed point and the robustness of the schemes to noise are determi ned. It is found that the domain of control for the two schemes is sim ilar for appropriately matched values of R and N, and that N-time-dela y autosynchronization tends to be less sensitive to noise.