125.

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		2.5.14— New Methods: Average Direction of the Trajectories

		There is an ongoing effort to overcome the problems in the phase space analysis and to develop new methods.

		As a system evolves in time, the point that represents the state of the system moves through the phase space. The trail of this point through the phase space is called the trajectory. The trajectory forms the phase space set. A large amount of data is needed to produce a long enough trajectory to determine the fractal dimension of the phase space set. Instead of evaluating the fractal dimension of the phase space set, Kaplan and Glass suggested evaluating the motions through the phase space set. This can be done by evaluating the average directions of the trajectories that pass through each small region in the phase space.

		1— Random: Average Directional Vector Is Small

		When the trajectory is constructed from data that were generated by a random mechanism, then the motion of the point in phase space is an aimless random walk. Each time the point passes through a small region in phase space it will pass through going in a different direction. Thus the average of the directions will be small, because the motion in different directions will cancel out.

		2— Deterministic: Average Directional Vector Is Large

		When the trajectory is constructed from data that were generated by a deterministic mechanism, then the motion of the point in phase space is highly organized, forming an attractor. Each time the point passes through a small region in phase space it will pass through going in approximately the same direction. Thus the average of the directions will be large, because the motions are in the same direction and so add together.

 

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