Lorenz Attractor Simulation

Edward Lorenz found these three equations in 1963 while cutting a weather model down to its bones. Nothing in them is random, and nothing is approximate: give the same three starting numbers twice and you get the same path twice, forever. Yet change the eighth decimal place of one of them and within about twenty time units the two paths are on opposite wings of the butterfly. That is sensitive dependence on initial conditions, and it is a property of the equations, not a flaw in the computer. It also has an off switch: the wing centres sit at ±sqrt(beta(rho - 1)) and stay stable until rho passes sigma(sigma + beta + 3)/(sigma - beta - 1), which for Lorenz's own numbers is 470/19, about 24.74. Below that, both twins settle on the same point and the future is perfectly predictable. Above it there is nowhere left to settle. (Strictly, a strange attractor already coexists with those stable points from about rho = 24.06, so in the narrow window between the two values the ending depends on where you start.) Students can change how fast x chases y and how hard it is driven and compare what happens. Use it for math inquiry with High School on a projector or student device.

Standards: Common Core CCSS.MATH.CONTENT.HSF.IF.B.4

• Run it at Lorenz's own numbers and watch the two coloured heads: they travel as one for about five seconds, then the stage rings the moment they part and stamps the time on it
• Pull rho down to 15 and release again — both twins now spiral into the same wing centre, and the gap readout shrinks instead of growing
• Pull rho below 1 and watch every path slide into the origin and stop: no wings exist at all
• Set the twin's head start to its smallest value and time how much longer they stay together — every extra decimal place buys only about a second
• Drag the starting point close to the vertical axis, where x and y are nearly zero, and watch the path slide a long way down before it is thrown out onto a wing: the axis is a resting place nothing can quite land on

1. 1Predict how changing how fast x chases y will affect the wing centre distance, then test the prediction.
2. 2Use evidence from the live calculation and visual to explain the result.

Questions teachers ask