High-Precision Measurement of the Feigenbaum Constant in an R-L-D Circuit

This repository contains the pdf file, Python simulation script, and final report for the research paper: "High-Precision Experimental Determination and Theoretical Modeling of the Feigenbaum Constant in a Driven Nonlinear R-L-D Oscillator."

This project presents an experimental and theoretical investigation into the period-doubling route to chaos. It goes beyond a standard replication by validating experimental results with a high-fidelity numerical simulation written in Python, achieving excellent agreement with theory.

Key Result: Simulation vs. Experiment

The primary result of this work is the close agreement between the experimental data and a numerical simulation based on a first-principles model of the circuit. The simulation, run on multiple CPU cores for efficiency, successfully reproduces the period-doubling cascade and the onset of chaos observed in the laboratory.

Figure 1: The final bifurcation diagram plotting the results of the numerical simulation (black points) against the bifurcation points measured experimentally (red dashed lines).

Circuit Diagram and Physical Model

The experiment is based on a driven series R-L-D circuit, as detailed below. The non-linearity of the diode is the key to observing the period-doubling route to chaos. The diagram also critically includes real-world parameters like the function generator's internal resistance and the oscilloscope's input capacitance.

The physical and electrical characteristics of the components shown are:

• Function Generator: The driving force for the circuit, providing a sinusoidal AC signal. It has a significant internal resistance of 50 Ω, which must be accounted for in any accurate model as it affects the total voltage delivered to the external circuit.
• Series Components:
  • A discrete Resistor of 5 Ω.
  • An Inductor of 221 µH.
• Nonlinear Element (Diode): A forward-biased diode acts as the essential nonlinear component. Its inherent junction capacitance is specified as 15 pF. This capacitance is what allows it to function within a resonant system at high frequencies.
• Measurement and Parasitic Effects:
  • The primary output variable, the Voltage of the Diode, is measured using an oscilloscope.
  • Crucially, the oscilloscope is not a perfect measurement device and introduces its own parasitic input capacitance, shown here as 18 pF. This capacitance is in parallel with the diode's own capacitance.
  • Therefore, the total effective capacitance at this node, which must be used for an accurate simulation, is the sum of the two: 15 pF (diode) + 18 pF (scope) = 33 pF.

Key Features

• Detailed Theoretical Model: Derives the circuit's governing second-order nonlinear differential equation using the Shockley diode equation.
• High-Precision Experimental Data: Utilizes data from an automated laboratory setup for precise measurement of bifurcation points.
• Parallelized Python Simulation: Includes a heavily commented Python script that uses multiprocessing and scipy to numerically solve the system's ODE and generate the bifurcation diagram efficiently.
• Rigorous Uncertainty Analysis: Calculates the final value of Feigenbaum's constant, $\delta$, with a statistically sound uncertainty.

Execute the Script**

Run the script from your terminal. It will automatically use all available CPU cores to speed up the computation and show a progress bar.

python bifurcation.py

The script will take a few minutes to run. When it's finished, it will print the total simulation time and display the final plot on your screen. The plot will also be saved as simulation_vs_experiment.png.

Abstract

The universality of the period-doubling route to chaos, characterized by the Feigenbaum constant $\delta$, is a cornerstone of nonlinear dynamics. The universality of the period-doubling route to chaos, characterized by the Feigenbaum constant δ, is a cornerstone of nonlinear dynamics. While the Resistor-Inductor-Diode (R-L-D) circuit is a canonical system for demonstrating this phenomenon, previous experimental realizations have often lacked rigorous theoretical modeling and comprehensive uncertainty analysis. This work presents a high-precision experimental determinationofδusinganautomated, computer-controlledR-L-D circuit. We develop a first-principles theoretical model based on the Shockley equation and the diode’s nonlinear junction capacitance to derive the system’s governing second-order non- linear differential equation. The bifurcation points leading to chaos are measured with high resolution, yielding a bifurcation diagram and power spectra that confirm the period-doubling cascade. From the measured bifurcation voltages, we calculate Feigenbaum’s first constant to be δ = 4.67 ± 0.08, a value in excellent agreement with the accepted value of 4.669. The analysis demonstrates that deviations in higher-order bifurcations can be qualitatively explained by non-ideal component behavior, highlighting the synergy between precision measurement and robust theoretical modeling in the study of complex systems.

Citation

If you use this work for reference, please cite it as follows:

@misc{Leung2025Feigenbaum,
  author       = {Leung, Shek Lun Alan},
  title        = {High-Precision Experimental Determination and Theoretical Modeling of the Feigenbaum Constant in a Driven Nonlinear R-L-D Oscillator},
  year         = {2025},
  publisher    = {GitHub},
  journal      = {GitHub repository},
  howpublished = {\url{https://github.com/alanspace/IPT}}
}

License

This project is licensed under the MIT License. See the LICENSE file for details.