[Paper Review] Electronic Implementation of the Mackey-Glass Delayed Model

Pablo Amil, Cécilia Cabeza+1 more|arXiv (Cornell University)|Aug 21, 2014

Nonlinear Dynamics and Pattern Formation9 references7 citations

TL;DR

This paper presents a novel electronic circuit implementation of the Mackey-Glass delayed differential equation using a switched-capacitor filter for precise time delay and a multiplier-based nonlinear function block. The circuit's discrete-time evolution equation closely matches the original continuous-time model, with experimental data, simulations, and the effective circuit equation showing excellent agreement across periodic, period-doubling, and chaotic regimes for varying time delays.

ABSTRACT

The celebrated Mackey-Glass model describes the dynamics of physiological extit{delayed} systems in which the actual evolution depends on the values of the variables at some extit{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.

Motivation & Objective

• To develop a stable, accurate electronic analog of the Mackey-Glass delayed differential equation for experimental study of time-delayed dynamics.
• To overcome limitations in prior implementations by precisely modeling the nonlinear feedback function and the time delay using analog components.
• To derive an exact discrete-time evolution equation for the circuit that accurately reflects the behavior of the original continuous-time system.
• To validate the circuit through direct comparison of experimental data, numerical simulations of the original equation, and simulations of the derived circuit equation.
• To enable deeper experimental investigation of complex dynamics such as chaos and period-doubling in time-delayed systems.

Proposed method

• The Mackey-Glass equation is transformed into a dimensionless form with reduced parameters using variable scaling for $ x = P/\Theta $ and $ t' = \gamma t $.
• A switched-capacitor filter is used to implement the time delay block, providing a stable, tunable delay with a transfer function that enables exact derivation of the circuit's evolution equation.
• The nonlinear term $ \alpha x_\Gamma / (1 + x_\Gamma^n) $ is implemented using analog multipliers and dividers to ensure precise control of the exponent $ n $.
• The circuit is designed with a specific block ordering—nonlinearity before the delay—to minimize noise and improve signal fidelity in experimental measurements.
• The effective evolution of the circuit is derived as a discrete-time map, resulting in the equation $ v_{j+1} = v_j e^{-T_s / (RC)} + (1 - e^{-T_s / (RC)}) \beta \frac{v_{j-N+1}}{\theta^n + v_{j-N+1}^n} $.
• The system is simulated using a 5th-order Runge-Kutta method for the original equation and a discrete map for the circuit model, with experimental data collected from the physical circuit.

Experimental results

Research questions

• RQ1 Can a stable and accurate electronic circuit be built to emulate the Mackey-Glass delayed differential equation with controllable time delay and nonlinearity?
• RQ2 How closely does the discrete-time evolution of the electronic circuit match the continuous-time dynamics of the original Mackey-Glass equation?
• RQ3 To what extent do experimental measurements of the circuit reproduce the bifurcation structure and chaotic behavior observed in numerical simulations?
• RQ4 What role does the ordering of the nonlinear and delay blocks play in minimizing noise and improving signal quality in the experimental setup?
• RQ5 Can the derived discrete-time equation of the circuit serve as a reliable model for predicting the behavior of the physical system?

Key findings

• The bifurcation diagrams from the original continuous-time equation, the discrete-time effective circuit equation, and experimental data all show excellent agreement, with identical transitions between fixed points, periodic windows, and chaotic regimes as the time delay $ \Gamma $ is varied.
• For $ \Gamma = 3 $, the system exhibits a simple periodic oscillation with one peak per cycle, confirmed in both simulations and experiments.
• At $ \Gamma = 5 $, a period-doubling bifurcation leads to a waveform with two peaks per cycle, observed consistently across simulations and experimental data.
• For $ \Gamma = 7 $ and $ \Gamma = 8.5 $, the system displays increasingly complex periodic and eventually chaotic behavior, with experimental waveforms matching simulated time series in shape and dynamics.
• The circuit's discrete-time evolution equation, derived from the switched-capacitor filter's transfer function, accurately models the system's behavior, especially with $ N = 1194 $ sampling points in the delay line.
• The experimental data show significantly reduced noise levels due to the strategic ordering of the nonlinear function before the delay block, enhancing signal clarity and measurement fidelity.