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For Educators · Chaos Explorer

Chaos Explorer for a water rocket

Built for educators teaching it to a class. Drop a live demo into a lecture or assign it as a shareable link — no lab installs. Simulate a water rocket live below — adjust the inputs and watch it respond, right in your browser.

Double Pendulum

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Double Pendulum StudioLive

chaotic dynamics · RK4

Controls

Speed1×

Tiny changes in the start explode into totally different motion — deterministic chaos. Drag either bob to set a new starting position; it pauses playback automatically.

Presets

Lower bob mass

Gravity g

▶ Run in Python

Data Inspector

IntegratorRK4

SystemChaotic

DOF2

Lower bob mass10

Gravity g9.8

Governing equation

\ddot{θ}_{1} \ddot{θ}_{2} = \frac{- g ( 2 m _{1} + m _{2} ) sin θ _{1} - m _{2} g sin ( θ _{1} - 2 θ _{2} ) - 2 m _{2} sin ( θ _{1} - θ _{2} ) ( θ ˙ _{2}^{2} L _{2} + θ ˙ _{1}^{2} L _{1} cos ( θ _{1} - θ _{2} ) )}{L _{1} ( 2 m _{1} + m _{2} - m _{2} cos ( 2 θ _{1} - 2 θ _{2} ) )} = \frac{2 sin ( θ _{1} - θ _{2} ) ( θ ˙ _{1}^{2} L _{1} ( m _{1} + m _{2} ) + g ( m _{1} + m _{2} ) cos θ _{1} + θ ˙ _{2}^{2} L _{2} m _{2} cos ( θ _{1} - θ _{2} ) )}{L _{2} ( 2 m _{1} + m _{2} - m _{2} cos ( 2 θ _{1} - 2 θ _{2} ) )} m_{1} = 10, m_{2} = 10, L_{1} = L_{2} = 1.2, g = 9.8

Reading this result: Starting both arms at 90° (π/2) with lower-bob mass 10 and g 9.8 puts the pendulum firmly in its chaotic regime. Unlike a small-angle swing, which would be near-periodic, here trajectories that differ by 0.02 rad diverge exponentially — a positive Lyapunov exponent.

Runs locally in your browser — free forever. Scale to the cloud when reality gets heavy.

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Attractors

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Strange Attractor GalleryLive

3D chaos · drag to orbit

Controls

Five famous strange attractors in 3D. Tune the Lorenz constants, then drag to orbit around the chaos.

Presets

σ (sigma)

ρ (rho)

β (beta)

Integration points

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Data Inspector

AttractorLorenz

Points19,799

ρ (rho)28

TypeStrange / chaotic

Governing equation

\overset{x}{˙} \overset{y}{˙} \overset{z}{˙} = σ (y - x) = 10 (y - x) = x (ρ - z) - y = x (28 - z) - y = x y - β z = x y - 2.667 z

Reading this result: Classic Lorenz (ρ=28, σ=10): the trajectory orbits two lobes forever without ever repeating. Sensitive dependence on initial conditions means the tiniest nudge changes everything — the butterfly effect.

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Bifurcation

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Bifurcation DiagramLive

logistic map · route to chaos

Controls

The logistic map xₙ₊₁ = r·xₙ(1−xₙ) doubles its period again and again as r grows, then dissolves into chaos near r ≈ 3.57 — with windows of order inside.

Presets

r min

r max

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Data Inspector

Mapr·x(1−x)

Chaos onsetr ≈ 3.57

Feigenbaum δ4.669

Governing equation

x_{n + 1} = r x_{n} (1 - x_{n}), r \in [2.5, 4]

Reading this result: Past r ≈ 3.57 the branches blur into chaos, yet pale vertical windows remain where periodic order suddenly returns.

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Frequently asked questions

Is this good for educators?: Yes — this version of "Chaos Explorer for a water rocket" is framed for educators teaching it to a class. Drop a live demo into a lecture or assign it as a shareable link — no lab installs.
Do I need to install anything?: No. It runs in any modern browser, free, with no account required.