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MathidiaMathematics Academy
Dynamical systems
Lesson 35 of 75
Review

Chaos and discrete dynamics

Objective. Analyse sensitive iteration and invariant structure.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Chaos in discrete dynamics describes iterated maps xₙ₊₁ = f(xₙ) whose long-term behaviour is extremely sensitive to initial conditions, despite the rule f being entirely deterministic. The logistic map xₙ₊₁ = rxₙ(1−xₙ) exhibits a period-doubling route to chaos as r increases, moving from a single stable fixed point through cycles of growing period to aperiodic, seemingly random orbits for r near 4. Sensitivity is measured by the Lyapunov exponent, which is positive exactly when nearby orbits separate exponentially fast under iteration. Example: at r = 4, two initial points differing by 10⁻¹⁰ can diverge to produce completely different sequences within a few dozen iterations.

Worked example

For the logistic map xₙ₊₁ = rxₙ(1−xₙ) with r = 4 and x₀ = 0.1, compute x₂ to illustrate the discrete dynamics before chaos sets in.

1 … 4
Illustrated card showing 1 … 4
  1. 1Compute x₁ = 4(0.1)(1−0.1) = 4(0.1)(0.9) = 0.36.
  2. 2Compute x₂ = 4x₁(1−x₁) = 4(0.36)(0.64).
  3. 3Evaluate: 4 × 0.36 = 1.44, then 1.44 × 0.64 = 0.9216.
  4. 4So x₂ = 0.9216, obtained purely from the deterministic rule applied twice.

Answer: x₂ = 0.9216

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Common mistake. Students often think chaotic behaviour implies randomness in the equations themselves; chaos arises from deterministic rules, with unpredictability coming purely from sensitive dependence on initial conditions, not from any random input.

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