59.

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2.3.1—
Lorenz System:
Physical Description
We use the Lorenz system to introduce the concept of sensitivity to initial conditions.
Lorenz studied the properties of a simplified model of the motion of the air in the atmosphere. The air is heated from below and cooled from the top. Hot air rises and cold air falls. The air rises and falls along the opposite edges of long cylinders. It is as if the cylinders of air are rotating. At the beginning one cylinder is rotating clockwise. As it turns, it brings hot air up from the bottom into the cold air at the top. It also brings cold air down from the top into the hot air at the bottom. Thus it mixes the air, reducing the temperature difference in the air. This temperature difference is the force driving the motion of the air. Hence, as the cylinder turns, it reduces the force driving its own motion. Thus the motion of the cylinder slows down and stops. However, without the motion of the air, the heat imposed from the bottom and the cold imposed from the top soon builds up the temperature difference in the air once again. Thus the cylinder starts to rotate again. Only now, it rotates in the opposite direction, counterclockwise. It rotates counterclockwise for a while. Again, as it rotates it reduces the force driving its own motion. It stops, the temperature difference builds up, and now it switches directions again, and rotates clockwise.
In summary, the Lorenz system consists of the motion of air along cylinders that rotate first clockwise, then counterclockwise, then clockwise, and keep switching from one direction to the other.
This system can be approximated by 3 equations and 3 independent variables. We will consider only the variable X. The value of X is the amount of angular velocity. That is, when X>0, a cylinder of air rotates clockwise; and when X<0, a cylinder of air rotates counterclockwise.
A phase space can be constructed from the variables X(t), Y(t), and Z(t). At time t, the system is represented by a point in the phase space with coordinates X(t), Y(t). and Z(t).

 
[Cover] [Abbreviated Contents] [Contents] [Index]


Fractals and Chaos Simplified for the Life Sciences
Fractals and Chaos Simplified for the Life Sciences
ISBN: 0195120248
EAN: 2147483647
Year: 2005
Pages: 261

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