Thursday, January 22, 2015

The Lorenz Attractor and Sensitivity to Initial Conditions

    I have previously posted on the Lorenz Attractor in Scratch (see:     http://www.scratch-blog.com/2013/01/the-lorenz-attractor.html).
   In that post I described the equations that define the attractor and that the attractor models a simple convection cell powered by a source of heat. 
    The importance of the Lorenz Attractor is that it is a nonlinear system that exhibits a sensitivity to initial conditions. In a linear system, small changes in inputs produce small changes in outputs.

    In this project, two points that start very close together—small changes in inputs—quickly diverge—large changes in outputs.
   At first, the two “particles” (each representing a slightly different starting state of the convection cell at time t) remain close together. But, after approximately 1400 iterations, the two points start to separate (see small circles in above graphic). 
   This behavior demonstrates how a very slight difference in the initial positions (x, y, z coordinates) of the two particles quickly lead to two different states of the cell.
   In the display, the two lobes can be thought of as representing a rotation of the convection dell in one direction and the other lobe as the rotation of the cell in the opposite direction.
   You can view and download this project by clicking on the following link.
http://scratch.mit.edu/projects/42648660/
   I have written a more detailed description of this project (including a sample of Dr. Lorenz's original data) that can be had in PDF form by emailing me at grandadscience@gmail.com.

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