Experiment 1
Keep initial states fixed and change r from 3.9 to 2.5. Predict the long-run gap.
Check the prediction
Both approach the attracting fixed point 0.6, so the gap shrinks. Nonlinearity does not automatically imply chaos.
Nonlinearity and chaos
A fixed rule can defeat long forecasts.
Iterate the logistic map, distinguish deterministic sensitivity from noise, and avoid declaring chaos from one irregular plot.
Start with: Repeated multiplication. A fixed point reproduces its own value under an update; a derivative measures local sensitivity.
01 · Commit to a prediction
What do you expect?
No. The rule is deterministic, but nearby initial states can separate strongly after repeated updates. That does not mean every nearby pair separates on every step.
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02 · Change an assumption
Predict. Change. Explain.
Before changing a control, say what should move and why. Start with the experiments below. Reset restores the starting model; it preserves your written responses.
Experiment 2
At r = 3.9, set δ = 0. Explain what this checks.
Check the prediction
Both computer trajectories are identical. The separation came from different starting states, not injected random noise.
0 to 4 · step 0.05
0 to 0.99 · step 0.01
1 to 60 · step 1
Calculated model output. The table gives the same values. Displayed values are rounded; calculations keep full precision.
Read the plotted data as a table
03 · Connect the mechanism
From the picture to the quantities.
The logistic map folds the interval back into itself. At some parameters, repeated stretching and folding can amplify uncertainty in the initial state. There is no random-number generator in this page. Its floating-point calculation is an approximation to real arithmetic, so late digits should not be mistaken for an exact long-term forecast.
xₜ₊₁ = r xₜ(1−xₜ)
Small perturbation: δₜ₊₁ ≈ r(1−2xₜ)δₜ
Nonzero fixed point: x* = 1−1/r (r > 1)
Small perturbation: δₜ₊₁ ≈ r(1−2xₜ)δₜ
Nonzero fixed point: x* = 1−1/r (r > 1)
x is a dimensionless state in [0,1]. r is dimensionless, 0 to 4. x₀ is 0 to 0.99 here, leaving room for δ ≤ 0.001. t counts updates; lines connect discrete points. δ is an initial displacement, not fresh measurement noise. The perturbation approximation applies only while the difference is small.
Worked example
At r = 4, x₀ = 0.2 gives x₁ = 0.64 and x₂ = 0.9216. Starting at 0.201 gives x₁ = 0.642396, so the first difference is 0.002396. The local prediction is 4(1−0.4) × 0.001 = 0.0024; its slight error is the omitted quadratic term.
Assumptions and limits
The interval [0,1] is invariant for 0 ≤ r ≤ 4. A nonzero fixed point attracts nearby states for 1 < r < 3. Other parameters can yield cycles or chaos, including periodic windows amid chaotic ranges. A finite irregular trace or a separating pair is not a proof of chaos. This discrete map is not the continuous logistic growth equation and is not a fitted forecast of a population.
The annotated sources distinguish established results from this lesson’s original examples.
04 · Follow the structure
Where else does this apply?
Forecast horizons
Sensitive dynamics can amplify small uncertainty in a measured initial condition.
Boundary: Real weather has many coupled variables; this map gives no weather forecast horizon.
Iterative numerical procedures
A deterministic iteration can alternate, converge or fail as a parameter changes.
Boundary: Sensitivity in one recurrence does not diagnose every algorithm or every nonlinear system.
05 · Retrieve without hints
Close the explanation. Try a new case.
Write an answer before opening its feedback. Later, return directly here without rereading above. Recognition, explanation and transfer are separate outcomes. No page action or answer reveal measures mastery.
Recognition and explanation
Reveal reasoning and rubric
No. Determinism specifies a unique next state given exact inputs. Limited measurements and arithmetic can prevent accurate long forecasts in sensitive regimes.
Self-check: Separate the rule from knowledge of its inputs.
Calculation
Reveal reasoning and rubric
Both are fixed points: 2 × 0.5 × 0.5 = 0.5 and 2 × 0 × 1 = 0. But a small positive perturbation near zero grows initially, while perturbations near 0.5 contract.
Self-check: Check both updates and distinguish existence from stability.
Novel transfer
Reveal reasoning and rubric
No. Noise, changing inputs, transients and deterministic dynamics can all look irregular. You need a justified model, repeated measurements and sensitivity or other dynamical evidence.
Self-check: Give at least one alternative explanation and a way to discriminate it.
On a later day, try again and record actual evidence in the curriculum. A later unaided explanation and a fresh transfer problem give stronger evidence than immediate familiarity. No reminder is scheduled.
Sources & scope
What supports the lesson?
- MIT 8.09, Chapter 7: Chaos and Non-Linear Dynamics. Logistic map, fixed-point stability and sensitive dependence.
Original teaching examples. Reference links need a connection; the lesson itself does not. Built 2026-10-11. Learner understanding is not assessed.