Gradient descent as an iterated map: bifurcations, chaos and fractal basin boundaries in neural-network training

Supplementary material for the review article "Gradient descent as an iterated map: bifurcations, chaos and fractal basin boundaries in neural-network training" by Ali Tozar, submitted to Chaos, Solitons & Fractals (2026). The article reads large-step gradient descent (GD) as the iteration of a one-parameter map and checks a few of its statements directly on the two smallest models where they can be solved: the double-well loss L(x) = (x^2 - 1)^2/4, whose GD map is the cubic g_a(x) = (1+a)x - a x^3 with a = eta, and the two-factor loss L(u,v) = (uv - 1)^2/2. All computations in this record concern these two reduced models. None of them is an experiment on a trained neural network. CONTENTS- code/: Python scripts (NumPy, Matplotlib). v01, v06, v09, v10, v11 are exact rational-arithmetic checks of the thresholds, conjugacies (Chen et al. cubic, Rogers-Whitley map, Kalra et al. map) and flip-criticality formulas, with short numerical confirmations. v02-v05 are the numerical experiments: period-doubling cascade and Feigenbaum ratios, the two-factor model, catapult time series, transversal Lyapunov exponent, riddling test, and uncertainty exponents of basin boundaries. figdata.py regenerates all figure data.- data/, figures/data, figures/raster: intermediate arrays, curve data and basin images used in the five figures.- figures/: the five figures as printed (PDF) and their LaTeX/pgfplots sources.- expected_output/: reference logs of every script.- README.md: maps each number in the article to the script that produces it, with its evidence level (exact, numerical, or drawn only), explains how to read the output, and states what the computations do not establish.- run_all.sh (quick / medium / full), build_figures.sh, requirements.txt, CITATION.cff, SHA256SUMS.txt. QUICK STARTpip install -r requirements.txtbash run_all.sh quick (under 10 seconds: all exact checks)bash run_all.sh medium (a few minutes: cascade, two-factor model, riddling, K2 focus)bash run_all.sh full (10-30 minutes: adds uncertainty exponents and the figure-data chain) LIMITATIONS (also stated in Section 8 of the article)Reduced models only; double-precision arithmetic with no higher-precision re-classification of boundary points; "which minimum" classes are counted before quotienting by the model's symmetry; outcomes are decided after a finite iteration cutoff; fitted exponents carry statistical errors only. Tested with Python 3.13, NumPy 2.5.3, Matplotlib 3.11.2 on Linux. Code: MIT license. Data, figures and documentation: CC BY 4.0. Article DOI: to be added on acceptance.