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

> Explore chaos: logistic map cobwebs and zoomable bifurcation diagram, Lyapunov exponent, Feigenbaum constant, 3D Lorenz attractor and Poincaré sections.

- URL: https://qilingtools.com/en/physics/chaos
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Four tabs look at chaos from different angles. The logistic map tab draws a cobweb diagram, a time series and a bifurcation diagram computed column by column that you can zoom into by dragging a box. Below it sits the λ(r) curve on the same axis, and the tab estimates the Feigenbaum constants δ and α from superstable parameters. The Lorenz / Rössler tab integrates two orbits 10⁻⁸ apart with RK4 and shows them peel apart in a rotatable 3D view. The driven pendulum tab shows the phase portrait, Poincaré section and period-doubling diagram, and the sensitivity tab fits the exponential divergence on a log plot to estimate how far ahead you can predict.

## How to use

1. On 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. On the Lorenz tab, change σ, ρ, β or pick a preset, drag to rotate, and watch when the two orbits separate
3. On 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. On the sensitivity tab, change the initial offset d₀ and read off the fitted Lyapunov exponent and the prediction horizon

## Features

- Cobweb diagram, time series, attractor period and Lyapunov exponent update live
- Progressively rendered bifurcation diagram with box zoom, click-to-set r, and one-click views of the period-3 window and self-similarity
- Feigenbaum δ ≈ 4.669 and α ≈ 2.503 estimated from superstable parameters
- Rotatable 3D Lorenz and Rössler attractors, a two-orbit butterfly effect and the Benettin λ_max
- Driven damped pendulum phase portrait, Poincaré section and period-doubling diagram
- Exponential divergence fit on a log scale with a prediction-horizon estimate

## 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.

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