Lorenz Attractor

Lorenz Attractor
Fields: Dynamic Systems and Fractals
Image Created By: Aaron A. Aaronson
Website: Math Art

Lorenz Attractor

The Lorenz Attractor is a 3-dimensional fractal structure generated by a set of 3 ordinary differential equations.

Basic Description

A lorenz attractor is the graph produced by a simple system of ordinary differential equations. The equations are given as follows.

(1):$\frac{dx}{dt} = \sigma (y - x)$

(2):$\frac{dy}{dt} = x (\rho - z) - y$

(3):$\frac{dz}{dt} = xy - \beta z$

The values for $\sigma$, $\rho$, and $\beta$ are varied to produce different images. The values Lorenz used are $\sigma = 10$, $\rho = 28$, $\beta = 8/3$ to produce the image shown.

The applet above shows the complexity of the curve. [show more][hide]

Originally described by Edward Lorenz as equations that would model the unpredictable behavior inherent in weather, the equations model the motion of fluid simultaneously heated from the bottom and cooled from the top.

A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

Properties

The Lorenz differential equations are nonlinear and [[Deterministic s [...]

[Click to hide A More Mathematical Explanation]

Properties

The Lorenz differential equations are nonlinear and deterministic and chaotic, and also a strange attractor.

Chaotic

The equations, despite their simplicity, are remarkably sensitive to changes in $\sigma$, $\rho$, and $\beta$, and thus an early example of a chaotic system.

These figures, made using ρ=28, σ = 10 and β = 8/3, show the state of the two Lorenz Attractor, denoted by the blue and yellow lines, at time t=1,2,3. The two differ only by a margin of $10^{-5}$ in the initial x coordinate. While the two graphs seem identical at first, with the yellow almost totally covering the blue, by t=3, the divergence is apparent.

Fractal

The graph produced in this case is a fractal of Hausdorff dimension of 2.06 ± 0.01

Strange Attractor

Looking at the Lorenz attractor, we can see that the points will remain near the graph. We do not see points diverging away from the graph itself; instead they seem attracted to the center of the figure. Because of this, we can classify the Lorenz differential equations as an attractor.

Since the Lorenz attractor also has a non-integer dimension, it is a strange attractor by definition. This propensity to oscillate about the center

Teaching Materials

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Future Directions for this Page

-An explanation for why the attractor is a fractal and why it is a strange attractor. -A helper page for strange attractor would be good too. -Some applet that graphs the attractor and allows the user to fool around with starting position and the initial conditions.

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