61.

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		2.3.2— Lorenz System: Phase Space Set

		The phase space set of the Lorenz system does not fill the 3-dimensional phase space. The region occupied by the phase space set is the attractor. The fractal dimension of the attractor is approximately 2.03. Since this attractor has a fractal dimension that is not equal to an integer, this attractor is called a strange attractor.

		The low fractal dimension of the attractor reveals the fact that the Lorenz system is deterministic. The number 3 is the smallest integer greater than or equal to 2.03. This reveals that the Lorenz system consists of 3 equations with 3 independent variables.

		The phase space set of the Lorenz system looks like a butterfly.

		The right wing of the butterfly is in the part of the phase space where X>0. Thus all the points on this wing correspond to the cylinder rotating clockwise. The left wing of the butterfly is in the part of the phase space where X<0. Thus all the points on this wing correspond to the cylinder rotating counterclockwise.

		Each set of values of X(t), Y(t), and Z(t) measured for the physical system corresponds to one point in the phase space. As the physical system evolves in time, the point representing it moves through the phase space along the butterfly. For example, if the point starts on the right wing, X>0, the cylinder is rotating clockwise. The point loops around the right wing a number of times until it switches to the left wing. On the left wing, X<0, and the cylinder is rotating counterclockwise. The point loops around the left wing a number of times until it switches back to the right wing. Back on the right wing, X>0, and the cylinder is rotating clockwise again.

		The figure does not do justice to the fine structure of the attractor. The more the attractor is magnified, the more lines are revealed that trace out the motion of the point in phase space that represents the evolution of the state of the system in time. The attractor is fractal.

 

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