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		2.3.3— Lorenz System: Sensitivity to Initial Conditions

		Given a set of starting values for X, Y, and Z, we use the Lorenz equations to compute the values of the variables X(t), Y(t), and Z(t) at different times t. These starting values are called the initial conditions. We first run the computation with one set of starting values and then rerun it a second time with the starting value of X changed ever so slightly.

		1— Initial Condition: X=1

		First we compute the values of X(t) from the starting values X=1, Y= 1, and Z=1.

		The starting value of X= 1 is greater than zero. Thus the cylinder of air starts off rotating clockwise. After a short time X becomes less than zero and the cylinder is rotating counterclockwise. Then X becomes greater than zero and the cylinder is rotating clockwise again. As time goes by, X becomes less than or greater than zero, and so the rotation of the cylinder switches between counterclockwise and clockwise.

		2— Initial Condition: X=1.0001

		Then we compute the values of X(t) from the starting values X=1.0001, Y=1, and Z=1.

		That is, we now rerun the same computation, just starting the initial value of X ever so slightly differently. At the beginning, the values of X(t) closely follow the values in the first run. This is not surprising, because the initial values of X were so similar.

		However, after a while, at the same elapsed time, when X>0 in the first run, now X<0 in the second run. Even though the two runs were started with almost identical initial conditions, now the state of the system in the second run is completely different from the state of the system at the same time in the first run. That is, the cylinder of air that is rotating counterclockwise in the second run was rotating clockwise at the same time in the first run. This is called sensitivity to initial conditions.

		The Lorenz system amplifies small differences in the initial conditions into large differences in the values of the variables later on. The time it takes for these large differences to appear is equal to (1/l), where l is called the Liapunov exponent.

 

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