Bifurcation Diagrams: Publication-Ready Figures from ODEs

What it does

A bifurcation diagram claims two things: the branches exist where they are drawn, and they have the stability shown. This skill makes your agent compute the diagram from your model, check every branch, and draw it to publication standard. Unlike a brute-force parameter sweep, it keeps the unstable branches and pins down the points where stability changes. Give it your equations (in the message or in a Python file) and the parameter to vary, and it:

• Traces every branch of equilibria by pseudo-arclength continuation, so S-shaped curves pass through their folds as one branch instead of breaking off where a simple parameter sweep fails.
• Classifies stability from the Jacobian eigenvalues at every point (analytic Jacobian when the derivatives are easy, central differences otherwise).
• Locates folds (SN), branch points (BP) and Hopf points (H) by bisection on a test function: det J for real crossings, the bialternate product for Hopf, which also gives the onset frequency.
• Compares each point with its analytic value when one exists, and reports the error. In the demo run both points matched the exact values to about 1e-12.
• Checks stability by simulation: on every stable or unstable segment it perturbs a point, integrates, and confirms the trajectory returns or leaves as classified.
• Adds limit cycles past a Hopf point (cycle maximum and minimum from simulation, joined to the Hopf point as is standard for a supercritical Hopf, and said so in the caption), and can overlay slow-passage or hysteresis-sweep trajectories.
• Draws one consistent encoding: colour = branch family, solid = stable, dashed = unstable, every stability switch on its marked point, a two-part legend, exact journal column width, vector PDF plus 600-dpi PNG.
• Writes the caption draft and spec (fixed parameters, method, what the line styles mean, every located point) and a results JSON with each point, its state, residual, and analytic value and error when known. Then it looks at the rendered figure before handing it over.

No general plotting skill does this: the branches, their stability and the located points all come from the model, and each is checked.

What's included

• SKILL.md: the workflow (model, continuation, events, simulation checks, limit cycles, drawing rules, outputs, final look)
• scripts/bifurcation_plot.py: continuation, stability, event location, simulation checks, limit-cycle extents, drawing helpers and output writers, with a self-test
• scripts/plot_style.py: journal column widths, Okabe-Ito colours, embeddable fonts
• examples/: three worked models with known answers: two folds with hysteresis, a supercritical Hopf in the Brusselator, a pitchfork in a genetic toggle switch
• references/method.md: the algorithm, tolerances, failure modes and fixes, and a Methods-section template

Part of the Aurora Lab research pipeline

Works entirely on its own. With a research/ folder it writes figures to research/figures/ and located points to research/results/. Pairs well with Aurora Lab's figure skills (Paper Figures for plots from data, Figure Audit for reviewing finished figures), but needs none of them.