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		2.3.6— Lorenz System: A Strange Attractor

		1— Points Away From the Attractor Are Drawn Toward It

		The Lorenz attractor does not fill the 3-dimensional phase space. Only certain combinations of the values X, Y, and Z are present. We can push the system away from this set of combinations. For example, we can reach in and put all the hot air next to the cold plate on top and all the cold air next to the hot plate on the bottom. The system will then rapidly escape from this unnatural state back towards a more natural state. In the phase space, this unnatural state is represented by a point off the attractor. As the system evolves toward a more natural state, this point rapidly approaches the attractor.

		2— Points On the Attractor Pull Away from Each Other

		Sensitivity to initial conditions can be seen in the motion of points on the attractor.

		We start the system with a combination of values of X, Y, and Z that are part of the attractor. If we then rerun the system a second time with ever so slightly different values of these initial conditions, then the point representing the original run and the point representing the rerun will rapidly separate from each other. This is a reflection of the sensitivity to initial conditions. The two points separate exponentially fast from each other. For example, we could start both runs on the right wing where X>O, and the cylinder of air is rotating clockwise. After the same elapsed time, the point of the first run has switched to the left wing, where X<0, and the cylinder of air is rotating counterclockwise while the point of the second run is still on the right wing, where X>0, and the cylinder of air is rotating clockwise.

		Although they separate from each other, both points remain on the attractor. The attractor is of a limited size. Thus they cannot separate from each other indefinitely. What happens is that after a while they are folded back onto each other. They separate and they are folded back onto each other, again and again. Thus the finer we look, the more of these trails we find. The evolution of the system in time that is represented by the trail of the point in the phase space forms the fractal structure of the attractor.

 

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