From steady to chaotic: the logistic map's period doubling

AA HLAI HL Ambitious Also in: Differential equations, Numerical methods

Research question

For the population model xₙ₊₁ = r xₙ(1 − xₙ), at which values of r does the long-run behaviour change from a steady state to a 2-cycle, a 4-cycle and chaos, and can the first two changes be found exactly?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

A one-line model produces chaos. The first two transitions can be found with algebra and derivatives; the rest you can measure numerically and compare.

The mathematics you'll need

Iteration and cobweb diagrams
Fixed points by solving x = f(x)
Stability from |f′(x*)| < 1
2-cycles from f(f(x)) = x; the first two thresholds (r = 3 and r = 1 + √6) derived
Numerical estimates of later thresholds and their ratios

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed; iterate in a spreadsheet.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explore the iteration for several r.
Find fixed points and their stability.
Derive the 2-cycle and its threshold.
Estimate later thresholds numerically and look at their spacing.
Reflect on what chaos means for predicting a real population.

Pitfalls that cost marks

Plotting a bifurcation diagram without explaining it.
Mistaking transients for long-run behaviour.
Claiming exact values you only estimated.

Showing personal engagement

Predict the behaviour before iterating.
Find a window of order inside the chaos.
Connect it to a population you know about.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Iteration and cobweb diagrams; Fixed points by solving x = f(x)
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Iteration and cobweb diagrams; Fixed points by solving x = f(x)

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Generate the fractal or the iteration yourself, choose your own variation (a different rule, angle or starting value), and pursue a question you raised while exploring.

Reflection (D)

Reflect on the limits of the model: real coastlines and plants are only self-similar over a range of scales, and computer iterations carry rounding error. Say how that affects your numbers. For this idea, start with: plotting a bifurcation diagram without explaining it — say how it affects your answer.

Use of mathematics (E)

SL: Geometric sequences and series for lengths and areas, logarithms for dimension, and iteration of functions, each calculated and checked numerically.

HL: Proof of a limit or dimension, complex-number iteration with a derived condition, or analysis of fixed points and their stability using derivatives.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Estimate the ratio of successive threshold gaps and compare it with the published Feigenbaum constant.

Extending it for HL

Find fixed points and decide their stability with derivatives, or prove the limit you found numerically.

See a complete IA, marked

Our annotated exemplar Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) asks a different question, but shows how a complete fractals & chaos exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).