Big Maths Ideas · Beta

Chaos and the logistic map

Compare two nearby starting values under the same simple rule.

Compare the blue start x₀ with the red start x₀ + ε. Increase the step count to follow both.

x_{n+1}=r x_n(1-x_n)

Both sequences use r = 3.9. Compare how their small starting difference develops.

Blue start x₀: 0.4
Red start x₀ + ε: 0.4001
Latest blue value: 0.60924
Latest separation: 0.36496

Blue is x₀; red dashed is x₀ + ε. Some parameters settle or cycle, while others show sensitive behaviour. A finite, rounded computer orbit is an illustration, not a proof of chaos or a forecast of a real population.

Predict

Does every parameter r in the logistic map produce chaos?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Does every parameter r in the logistic map produce chaos?

$x_1=3(0.2)(1-0.2)$

Hint

Apply the same rule to both sequences at every step.

Reveal answer

Some choices settle to a fixed value or a cycle; sensitivity depends on the parameter and starting values.

$0.48$

Build an explanation

Hint 1

Apply the same rule to both sequences at every step.

Hint 2

Some choices settle to a fixed value or a cycle; sensitivity depends on the parameter and starting values.

Hint 3

Compare the result with the model, then explain why it occurs.

Worked example

This example uses fixed values, separate from the diagram controls.

Apply one step with r = 3 and x₀ = 0.2.

Use x₁ = r x₀(1 − x₀).
Substitute 3 × 0.2 × 0.8.
Multiply without rounding early.

Answer: x₁ = 0.48.

Connect this idea

Watch out: Some choices settle to a fixed value or a cycle; sensitivity depends on the parameter and starting values.