Chaos theory and the butterfly effect

Equipment

• 3D phase-space trajectory model (panel 1)
• Initial-separation δ and parameter ρ sliders
• Panel 2 time-separation plot

Procedure

1. Two nearby trajectories

   Set ρ = 28 (classic chaotic regime) and small δ (~0.1). Watch the two trajectories stay glued together for a stretch then suddenly peel onto different «wings» of the attractor — the signature of sensitivity to initial conditions.

2. Measure divergence rate vs δ

   Sweep δ through a few values (small, medium, large) and record when the two tracks become distinct in panel 2. Smaller δ only delays the split since separation grows exponentially as $d(t)\approx d_0 e^{\lambda t}$ — λ stays fixed while only the «threshold time» shifts.

3. Sweep ρ across the chaos boundary

   Lower ρ toward 20–24: the trajectory may settle onto a stable fixed point instead of chaos (the stability boundary sits near ρ ≈ 24.74). Compare the attractor shapes on both sides of the threshold — bifurcation as a parameter crosses a critical value is a central notion of chaos theory.

4. Estimate the Lyapunov exponent

   With δ = 0.1, read two times (t₁, t₂) and separations (d₁, d₂) from panel 2 in the log-linear growth region. Compute $\lambda\approx\ln(d_2/d_1)/(t_2-t_1)$; λ > 0 confirms chaos. Predict that λ measured with δ = 1 must be identical — check it.

Simulation

Experiment history

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