🦋Chaos Simulator - Logistic Map Bifurcation, Lorenz Attractor, Butterfly Effect

How to use Chaos & Strange Attractors

1. 1On the logistic map tab, drag r and watch the cobweb go from a fixed point to period 2, 4… and chaos; drag a box on the bifurcation diagram to zoom
2. 2On the Lorenz tab, change σ, ρ, β or pick a preset, drag to rotate, and watch when the two orbits separate
3. 3On the driven pendulum tab, step through the A presets and see the Poincaré section go from one dot to two to a fractal cloud
4. 4On the sensitivity tab, change the initial offset d₀ and read off the fitted Lyapunov exponent and the prediction horizon

FAQ

Is chaos the same as randomness?

No. Every system here is fully deterministic: the same starting point always gives the same result. Chaos means extreme sensitivity to initial conditions. The gap between two nearly identical starts grows like e^(λt), so any tiny measurement error eventually becomes as large as the system itself. Long-term behaviour is then unpredictable in practice and looks random.

How should I read the Lyapunov exponent?

It is the average exponential rate at which neighbouring orbits separate, per unit time or per step. λ < 0 means perturbations die out and the system settles on a periodic orbit; λ > 0 means chaos. For the logistic map at r = 4, λ = ln 2, so the error doubles every iteration. For the Lorenz system λ ≈ 0.906, so the error grows tenfold about every 2.5 time units.

Why does the Feigenbaum constant matter?

The gaps between successive period doublings shrink by a fixed ratio δ ≈ 4.6692, and the diagram is self-similar vertically by α ≈ 2.5029. Remarkably, every map with a single quadratic hump shares the same δ and α on its road to chaos. This universality has been seen in fluid convection, electronic circuits and lasers. The tool finds the superstable parameters Rₙ with Newton's method, which pins δ to six significant figures within a dozen levels.

Why is it called the butterfly effect?

In 1963 the meteorologist Edward Lorenz found that rounding the starting values of his simplified convection model (the Lorenz equations) to three decimals completely changed the forecast. He later asked whether a butterfly flapping its wings in Brazil could set off a tornado in Texas. On the Lorenz tab, even a 10⁻¹⁴ initial offset keeps the two orbits together for only about 35 time units.

What is a Poincaré section?

For a periodically driven system, recording the state (θ, ω) once per drive period gives a Poincaré section. A period-1 motion shows up as one dot and period 2 as two dots. Chaotic motion leaves a cloud with fine layered structure, a slice through the strange attractor. Raising the drive amplitude A shows the 1 → 2 → 4 → chaos route, just like the logistic map.