The most famous shape in chaos theory, spinning live in your browser. A single point traces the Lorenz equations forever, looping around two lobes into the iconic butterfly — never repeating, never crossing itself. Drag to rotate it in 3-D, turn the ρ dial, then split off a twin trajectory a millionth apart and watch them diverge. It's the 3-D companion to the bifurcation diagram and the double pendulum — the same butterfly effect, drawn in space.
Twin separation against time on a logarithmic axis. Exponential divergence — the butterfly effect — appears as a straight, climbing line, until the gap saturates at the size of the attractor. Turn on the twin (or the Twin-divergence preset) to plot it.
A millionth of a difference, and the future is unrecognisable — yet always the same butterfly. Edward Lorenz found this in 1963 when rounding a weather-model number from 0.506127 to 0.506 sent the forecast somewhere completely different. Run the twin here — a second trajectory started just 1e-3 away from the first, under the identical rule dx=σ(y−x), dy=x(ρ−z)−y, dz=xy−βz with σ=10, ρ=28, β=8/3. The two paths track together, then peel apart onto opposite wings. Neither ever leaves the attractor; neither ever repeats. Determinism does not mean predictability.
Reading the simulation
What the spinning trail, the labels, the readouts and the log plot are telling you.
1
The trail is the solved trajectory
The glowing head is the current state (x, y, z), printed live in the readout, and the fading trail is where it has just been — the solution of the equations as they are integrated forward. The system name and its parameters are labelled on the canvas; for Lorenz, the two spiral centers C− and C+ at the hearts of the lobes are marked, and the head loops around one, then the other, in an order that never repeats.
2
Bounded, but never repeating
However long you watch, the head stays on the same butterfly-shaped surface — the strange attractor — yet never lands on a point it has visited before and never crosses its own path. The Regime readout names what you have: turn ρ below about 24.74 and the trajectory spirals into a fixed point and the motion dies; above it, the same rule produces sustained chaos. Bounded forever, predictable never.
3
The twin exposes sensitive dependence
Switch on the twin and a second point starts just 1e-3 away under the identical rule. At first the two heads sit on top of each other; then the Twin separation readout climbs and the Divergence factor blows up as they peel onto opposite wings. The lower panel plots that gap on a logarithmic axis, so the exponential runaway reads as a straight climbing line before it saturates at the width of the attractor — the butterfly effect, drawn as a graph.
How It Works
Three equations, integrated step by step, tracing a shape that never repeats.
1
Integrate the equations
The state is a point (x, y, z). Three coupled rules — dx=σ(y−x), dy=x(ρ−z)−y, dz=xy−βz — say how it moves each instant. We advance it with fourth-order Runge–Kutta, a small, accurate time step repeated many times per frame, so the path stays true even as it stretches and folds. The Speed slider sets how many of those steps run per frame; the Trail length sets how much of the past stays drawn.
2
Fixed points, then chaos
The ρ dial is the control parameter. For ρ below 1 the only rest state is the origin and every path decays to it; between there and about 24.74 the flow spirals into one of two off-center fixed points. Push ρ past that threshold — the classic value is 28 — and those points turn unstable, leaving nowhere to settle: the point is trapped on the strange attractor, bounded, fractal and chaotic. Rössler and Aizawa reach chaos through the same idea with different rules.
3
Determinism without predictability
Every step is exactly computed, so the future is fixed by the present — yet the twin shows why that is not the same as being predictable. Two starts a millionth apart follow identical arithmetic and still end up on opposite wings, because errors grow exponentially. That is sensitive dependence on initial conditions: the reason weather can be modelled precisely and still not forecast far ahead.
What is the Lorenz attractor?
The Lorenz attractor is the set of trajectories traced out by the Lorenz equations, a simplified model of atmospheric convection published by meteorologist Edward Lorenz in 1963. The three equations dx=σ(y−x), dy=x(ρ−z)−y, dz=xy−βz, with the classic constants σ=10, ρ=28 and β=8/3, produce a path that loops endlessly around two spirals without ever repeating, forming the famous butterfly-shaped figure in three dimensions. This simulator integrates those equations with fourth-order Runge–Kutta and draws the result as a rotating, fading trail, so the two lobes of the butterfly fill in as you watch.
What is a strange attractor?
An attractor is the shape a system settles onto in the long run — a point, a loop, or a surface that nearby trajectories are drawn toward. A strange attractor is one with fractal structure on which the motion is chaotic: trajectories stay confined to a bounded region yet never repeat and are exquisitely sensitive to their starting point. The Lorenz attractor was the first strange attractor to be widely studied, which is why it became the emblem of chaos theory.
What is the butterfly effect?
The butterfly effect is sensitive dependence on initial conditions: in a chaotic system, an arbitrarily tiny change in the starting state grows exponentially until the outcome is completely different. Edward Lorenz found this while rounding a weather-model input from 0.506127 to 0.506, and the name comes from his talk asking whether a butterfly's wings in Brazil could set off a tornado in Texas. Turn on the twin trajectory here — started just 1e-3 away — and watch the two paths, identical at first, drift onto opposite wings of the attractor. The separation-vs-time panel plots the gap on a logarithmic axis, where the exponential runaway shows up as a straight, climbing line until it saturates at the size of the attractor.
Why does the trajectory never repeat or cross itself?
The Lorenz system is deterministic, so its future is fixed by its current position — which means if the path ever crossed itself it would be forced into a repeating loop. Because the motion is chaotic and bounded on a strange attractor of fractional dimension, it must thread endlessly through the same region without ever landing on a point it has visited before. In three dimensions the curve comes close to itself again and again but never truly intersects.
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Classroom activity
A ready-to-assign lab using the simulation above. Free to use — nothing to install or sign up for, and no student data leaves the browser.
High school, College10–20 minInteractive simulation
Learning objectives
Describe deterministic chaos and strange attractors.
Observe sensitive dependence on initial conditions.
Distinguish chaotic from random behavior.
Variables to change
Initial conditions
System parameters
Procedure
Run the system and watch the trajectory trace the two-lobed attractor.
Start two trajectories from nearly identical points and watch them diverge.
Let it run and confirm the path never exactly repeats yet stays bounded.
Observations
Describe the shape of the attractor and how quickly two close starts separate.
Questions
What is a strange attractor?
Is the Lorenz system random?
What is the "butterfly effect"?
Does the trajectory ever settle to a point or loop?
What did Lorenz originally model with this?
Explanation
The Lorenz system is deterministic yet chaotic: trajectories are drawn to a bounded, never-repeating "strange attractor," and nearby starts diverge exponentially (the butterfly effect). It arose from weather modeling and helped found chaos theory.
Answer key (for instructors)
1. A bounded set that a chaotic trajectory is drawn to but never repeats, with fractal structure.
2. No — it is deterministic; the same start gives the same path, but tiny differences grow exponentially.
3. Sensitive dependence on initial conditions — small changes lead to vastly different outcomes.
4. No — it stays bounded but never repeats, endlessly circling the two lobes unpredictably.
5. A simplified model of atmospheric convection (weather), which is why it launched chaos theory.
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