Chaos control in multistable delay-differential equations and their singular limit maps

The multistability exhibited by first-order delay-differential equations (DDE’s) at large delay-to-response ratios $R$ is useful for the design of dynamical memory devices. This paper first characterizes multistability in the Mackey-Glass DDE at large $R.\mathrm{}$ The extended control of its unstable periodic orbits (UPO’s), based on additional feedback terms evaluated at many times in the past, is then presented. The method enhances the control of UPO’s and of their harmonics. Further, the discrete-time map obtained in the singular perturbation limit of the controlled DDE is useful to characterize the range of parameters where this extended control occurs. Our paper then shows how this singular limit map leads to an improved method of controlling UPO’s in continuous-time difference equations and in discrete-time maps. The performance of the method in the general contexts of extended additive and parametric control is evaluated using the logistic map and the Mackey-Glass map. The applicability of the method is finally illustrated on the Nagumo-Sato discrete-time neural network model.