Dynamical invariants, multistability, controllability and synchronization in delay-differential and difference equations.

Abstract

We have studied the properties of nonlinear delay-differential equations (DDE's) that commonly appear in models of physiological, neural control and of laser cavities. These systems exhibit multistability at large delays which makes them attractive for information storage purposes. We have considered the storage in stable periodic orbits of the dynamics of recurrent circuits, involving spiking and non-spiking neurons, and in unstable periodic orbits stabilized by delayed feedback-based chaos-control techniques. The information is encoded into the initial function from which the DDE evolves into a periodic solution. The storage capacity is found to be proportional to the delay time of the feedback loop and to be related to the unstable modes in the linearized system in the absence of control. In the chaotic regime, a combination of stable and unstable modes is used to explain the oscillations in power spectra. The latter is used to estimate the attractor dimension and the metric entropy, which are more simple to calculate than with Lyapunov and dimension techniques, especially when the attractor is high-dimensional. The power spectra also provide analytical insight into multistability of delay-differential and difference equations, an area in which there are still few results. The singular perturbation limit of the DDE provides an improved method to control unstable periodic orbits in discrete-time maps, and in particular, in neural networks that are modeled by discrete-time dynamics. This limit also provides insight into controllability and synchronization of DDE's. This is applied to the context of secure communication, where an information-carrying signal is masked or modulated by a DDE broadband carrier. The results are relevant to the study of control systems, physiological and other, involving multiple feedback loops.