Strange Attractors

Strange attractors are graphs created by using a system of equations that when applied over and over can produce chaos. I have two examples here. The symmetric icons are also strange attractors. Both of the strange attractors on this page are fractal in nature, although not all are. In either applet, click anywhere to zoom in, and hold Shift and click to zoom out.

Lorenz Attractor

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a
b
c
dt 		x1
y1
z1
 x2
y2
z2

This strange attractor was named after Edward Lorenz, a meteorologist who discovered the butterfly effect. The equations for this attractor are:

x' = x - a * x * dt + a * y * dt
y' = y + b * x * dt - y * dt - z * x * dt
z' = z - c * z * dt + x * y * dt
Lorenz showed that the smallest number of linear differential equations that can produce chaos is three. As short as these equations are, they can't be solved with calculus. The shape of the attractor is controlled by the variables a, b, and c. If you decrease dt, the lines will be smoother, though it will take longer to draw. Dt must be a small number greater than zero.

There are two sets of values for the initial x, y, and z. One controls the purple line and the other controls the yellow line. They are currently set to be very close so that you can see the butterfly effect. Set the number of iterations to a small number, such as 100. The two paths will be nearly identical. (One will probably cover up the other, but you can zoom in to see that they are slightly different.) Increase the number of iterations. They will begin to move apart. Eventually, they will diverge completely and look nothing like eachother.

Henon's Attractor

This strange attractor can be made by making an ellipse, bending it into a horshoe, placing the horshoe over the original ellipse, bending the new shape into a horshoe, etc. In this way, the ellipse is folded into one infinitely long loop. Zoom in on any section of any line in this attractor and you will discover that the line you saw is, in fact, infinitely many lines that are close together. The easiest place to see this is the lower left-hand corner.

The equations for this attractor are:

x' = y + 1 + 1.4 x ^ 2
y' = .3 x

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