32.

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

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2.2.1—
Phase Space
About a hundred years ago the mathematician Poincaré showed how to transform a sequence of values in time into an object in space. This transformation replaces an analysis in time with an analysis in space. The space is called the phase space. The object in the phase space is called the phase space set. The act of transforming the sequence of values in time into the object in space is called an embedding.
This transformation is useful because certain properties of the data are easier to determine from the phase space set than from the original sequence of values in time.
For example, let's follow the location of a baseball in Fenway Park in Boston. The sequence of values in time of its distance left, right, and above home plate are given by X(t), Y(t), and Z(t). At a time t1 the location of the baseball is given by the values X1, Y1, and Z1. The point in a 3-dimensional phase space with coordinates X=X1, Y=Y1, and Z=Z1, corresponds to the location of the baseball at that time. As the baseball moves, the values X, Y, and Z change. These values correspond to the coordinates of the point in the phase space. Thus the point moves in the phase space. This moving point traces out an object in the phase space. This object is the phase space set.
The sequence of values, X(t), Y(t), and Z(t), is not limited to positions in space. For example, X(t) could be the value of the temperature, Y(t) the value of the potassium concentration, and Z(t) the value of the electrical voltage measured from a cell. At a time t1, these measurements have values X1, Y1, and Z1. The point in a 3-dimensional phase space with coordinates X=X1, Y=Y1, and Z=Z1 corresponds to the state of the cell at that time. As the state of the cell changes, the values measured for X, Y, and Z change. These values correspond to the coordinates of the point in the phase space. As the state of the cell changes, these values change, and the point representing the cell moves in the phase space. As it moves it traces out an object. This object is the phase space set.

 
[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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