What is a strange attractor?

The definition

Each attractor in Chaos Studies comes from three ordinary differential equations. Together they give a velocity at every point in space. A path is computed by starting somewhere and following that velocity forward in small steps; the app steps the Lorenz system every 0.007 time units.

From a wide range of starting points, the path is drawn onto the same set and stays there. That set is the attractor. A strange attractor adds two things: the path never closes into a loop, and the set it fills has a definite structure, which is what separates it from noise. The motion is bounded, so it never leaves a finite region, yet it never repeats.

The simulation above is the Lorenz attractor. Each new point is laid down red and cools to white as it ages, so the red end of the trail is where the system is now.

Why it is called strange

In this sense, strange means fractal. Zoom in on part of the attractor and you find layers, and each layer is made of more layers. Its dimension is not a whole number: the Lorenz attractor's is about 2.06, more than a surface and much less than a solid.

Chaotic is a separate property. It means that nearby starting points separate exponentially, so long-term prediction is impossible even though nothing in the system is random.

Attractors that are not strange

Not every attractor is strange. A pendulum with friction swings a little less each time and comes to rest hanging straight down. Every path spirals into that one state, so its attractor is a single point: a fixed point.

A limit cycle is a closed loop that nearby paths are drawn onto, so the system settles into the same repeating rhythm wherever it starts. A pendulum without friction is not an example. It swings forever, but a harder push gives it a bigger swing, so nothing draws it onto one particular loop.

A strange attractor also draws paths in, but onto a set where the motion never repeats. The Lorenz system has three fixed points, and at the classic parameters all three are unstable, so the path never comes to rest. It circles the two at (±8.49, ±8.49, 27) instead, switching between them without a pattern.

Sensitive dependence

Two points that start a hair apart soon end up in unrelated places, yet both trace the same figure. For the Lorenz attractor the gap grows by a factor of e about every 1.1 time units, the rate set by its largest Lyapunov exponent of about 0.906. Every chaotic system has at least one positive Lyapunov exponent.

This is the butterfly effect, named after Edward Lorenz's 1972 talk. The Lorenz page runs two trajectories that start 0.00001 apart and plots how fast they separate.

Deterministic is not random

A chaotic system is fully deterministic. Run it again from exactly the same starting point and you get exactly the same path. The unpredictability comes from the fact that no starting point can be measured or entered with infinite precision. Lorenz found this in 1961 when he restarted a weather simulation with 0.506 where the computer had stored 0.506127, and the new run soon bore no resemblance to the old one.

The individual path is unpredictable, but the shape is not. Start anywhere in the attractor's reach and the path traces the same figure. Chaos Studies shows both at once: every point follows its own chaotic path, and together thousands of them draw a stable shape.

How space shrinks onto the attractor

Divergence is the rate at which the flow shrinks a small volume of starting points. For the Lorenz system it is −(σ + 1 + β), about −13.67 everywhere, so any small volume shrinks by a factor of e^13.67 each time unit. That is how the whole of space collapses onto a thin, layered set.

Nosé-Hoover is the exception among the nine. Its divergence averages to zero, so it does not shrink volume on average, and strictly speaking it has no attractor. The field guide lists the divergence of all nine side by side.