Chaos Lab – Nonlinear Dynamics Playground

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Pendulum A Pendulum B (offset 1e-6 rad)

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Drag to rotate · wheel to zoom

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Click / drag to set r · wheel to zoom · double-click to reset

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trajectory strobe (Poincaré)

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Drag to pan · wheel to zoom · click sets Julia c · double-click zooms

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Concept

Two rigid arms (equal masses, unit lengths) obey Lagrange's equations; with δ = θ₁−θ₂ the angular accelerations are

θ̈₁ = [−3g sinθ₁ − g sin(θ₁−2θ₂) − 2 sinδ (ω₂² + ω₁² cosδ)] / (3 − cos 2δ)
θ̈₂ = [2 sinδ (2ω₁² + 2g cosθ₁ + ω₂² cosδ)] / (3 − cos 2δ)

Integrated with fixed-step RK4 (h = 2 ms). Energy E = ω₁² + ½ω₂² + ω₁ω₂cosδ − 2g cosθ₁ − g cosθ₂ is conserved; the drift chip shows RK4's tiny error.

Butterfly effect: separation grows like d(t) ≈ d₀ e^λt until it saturates at the size of the attractor. λ is the largest Lyapunov exponent, estimated live by the Benettin method: evolve a shadow pendulum, renormalise its distance to d₀ every 10 ms and average ln(d/d₀)/t.

Predictability horizon ≈ (1/λ) ln(Δ/d₀): cutting the initial error 1000× buys only a few seconds more.

Concept

Lorenz (1963) truncated convection to three ODEs:

ẋ = σ(y − x)
ẏ = x(ρ − z) − y
ż = xy − βz

For ρ > 1 two equilibria C± = (±√(β(ρ−1)), ±√(β(ρ−1)), ρ−1) exist; they become unstable at ρ ≈ 24.74 (for σ=10, β=8/3) and trajectories wander forever on a fractal strange attractor of dimension ≈ 2.06.

Largest Lyapunov exponent λ₁ ≈ 0.906 for the classic parameters: nearby trajectories separate by e^{0.906 t}. The three exponents sum to −(σ+1+β) = −13.67 (volume contraction).

Trajectories are coloured by their initial offset (10^−k in x); watch them agree, then part ways at a time ≈ ln(1/d₀)/λ.

Concept

The logistic map models a population with growth rate r and crowding. Fixed points: x* = 0 and x* = 1 − 1/r, stable while |f′(x*)| = |2 − r| < 1, i.e. r < 3.

At r = 3 the fixed point loses stability and a period-2 cycle is born; period doubling repeats faster and faster (r_n → r_∞ ≈ 3.5699), after which chaos reigns, punctuated by periodic windows (period 3 at r = 1+√8 ≈ 3.8284).

Lyapunov exponent λ(r) = lim (1/N) Σ ln|r(1 − 2x_n)|: negative on periodic windows, positive in chaos. At r = 4, λ = ln 2.

Cobweb: bounce between the parabola (apply f) and the diagonal (x_n+1 → next input).

Concept

Iterate

z_n+1 = z_n² + c

The Mandelbrot set is the set of c for which the orbit of z₀ = 0 stays bounded; the Julia set J_c is the boundary of the points z₀ that stay bounded for a fixed c. J_c is connected exactly when c is in the Mandelbrot set.

Smooth colouring: when |z| passes the bailout R, use the fractional iteration count ν = n + 1 − log₂(ln|z_n|) so colour bands blend continuously.

Deep zoom: double precision runs out near 10⁻¹⁵. This lab computes one reference orbit at the view centre in 260-bit integer arithmetic and iterates every pixel only as a small perturbation

δ_n+1 = 2Z_nδ_n + δ_n² + δc

with rebasing to avoid glitches. Mandelbrot zoom works to about 10⁵⁰; Julia mode is plain double precision (≈10¹³).

The Mandelbrot boundary has Hausdorff dimension 2; the cardioid and bulbs carry attracting cycles of period 1, 2, 3…, matching the logistic map's period doubling along the real axis.

Automaton

Rule number

Cell size

Wrap edgesGrid lines

Concept

Elementary automata update each cell from itself and its two neighbours: 2³ = 8 neighbourhoods → 256 rules; the rule number's binary digits are the outputs. Rule 30 looks random (used for random numbers), 90 draws the Sierpiński triangle, and Rule 110 is Turing-complete: gliders on a periodic background carry computation.

Conway's Game of Life (B3/S23): a dead cell with exactly 3 live neighbours is born; a live cell with 2 or 3 survives. From this: still lifes, oscillators, spaceships (glider, LWSS), guns and universal computation.

s′ = [n = 3] ∨ ([s = 1] ∧ [n = 2]) (n = live neighbours of 8)

Wolfram's classes: I fixed, II periodic, III chaotic (rule 30), IV complex edge-of-chaos (rule 110, Life). Cell colour in Life shows age: yellow newborn → blue old.