40.

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Page 138
2.2.5—
Constructing the Phase Space Set from a Sequence of Data in Time
1—
Experimental Measurement of All the Variables
The values of each of the relevant properties X, Y, Z, . . . are measured simultaneously over time. One point in the phase space is plotted for each set of measurements made at one time. The coordinates of each point are the values of each property X(t), Y(t), Z(t), . . . measured at time t.
2—
Experimental Measurement of One Variable and Takens
' Theorem
The values of one property X are measured over time. Takens' theorem says that the entire phase space set can be constructed from one independent variable. This remarkable procedure works because the variables are linked together by the relationship that produces the attractor.
One point in a phase space of dimension N is plotted for each measurement at time t. The coordinates of each point are X(t), X(t+Dt), X(t+2Dt), . . ., X(t+(N-1) Dt). The value Dt is called the lag. These lagged coordinates generate an N-dimensional phase space set.
(A "real" phase space would consist of the coordinates X(t), dX(t)/dt, d2X(t)/dt2 . . . . The coordinates X(t), X(t+Dt), X(t+2Dt), .. are a linear combination of the differences that approximate these derivatives. Thus Dt must be chosen so that the differences such as [X(t+Dt)-X(t)]/ Dt are a good approximation of the derivatives. The fractal dimension of the phase space set is invariant under a linear transformation of the coordinates. Thus the fractal dimension of the phase space set computed in this way is equal to the fractal dimension of the "real" phase space set.)
The fractal dimension of the phase space set cannot be larger than the embedding dimension of the space that we put it into. Your 3-dimensional body casts only a 2-dimensional shadow on a 2-dimensional surface. Thus the embedding must be repeated for increasing N and the "real" dimension of the phase space set is the limiting value found as the embedding dimension N is increased.

 
[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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