Exact Discrete-Time Implementation of the Mackey–Glass Delayed Model

Abstract

The celebrated Mackey-Glass model describes the dynamics of physiological delayed systems in which the actual evolution depends on the values of the variables at some previous times. This kind of systems are usually expressed by delayed differential equations which turn out to be infinite-dimensional. In this contribution, an electronic implementation mimicking the Mackey-Glass model is proposed. New approaches for both the nonlinear function and the delay block are made. Explicit equations for the actual evolution of the implementation are derived. Simulations of the original equation, the circuit equation, and experimental data show great concordance.

Organization and identification of solutions in the time-delayed Mackey-Glass model

“…33, and numerically, by using a discrete-time equation that approximates the exact solutions of the MG model and in particular, models the delay line in the electronic circuit, which is implemented via a linear array of capacitors (a Bucket Brigade Device, BBD). We have found an excellent agreement between observations of the electronic circuit and the simulations of the discrete-time MG model.…”

“…30 Several groups have proposed experimental implementations of the MG model via electronic circuits. 5,19,29,[31][32][33] An electronic implementation based on an analog delay line was considered in Refs. 19,28, and 31 to address the problem of synchronization and control of high dimensional chaos.…”

“…They also showed that a finite message can be stored in a periodic waveform and that its storage capability is enhanced by the control of unstable periodic orbits in the chaotic regime. The scope of the present work is to take advantage of a recently proposed electronic implementation 33 to investigate the coexistence of different, but very similar, high-dimensional periodic and aperiodic solutions.…”

“…Additional details about the pre-and post-amplification and the evaluation of the inputs and outputs of the delay and the nonlinear functions are described in Ref. 33.…”

Characterizing multistability regions in the parameter space of the Mackey-Glass delayed system

“…In addition to numerical studies, different experimental systems based on electronic implementations that mimic the MG model were proposed [13,14]. Due to the analytical and numerical difficulties, experimental results play a major role and contribute to validate theoretical models.…”