The Lorenz Model - Demo 4

Power Spectrum

Another way to look at the randomness is to ask about the spectral properties, i.e. how much of the variation is at each frequency. This is given by the power spectrum, which is roughly the squared magnitude of the Fourier transform of the signal. A periodic signal, such as a pure musical tone, would give a series of spikes in the ideal power spectrum at the fundamental frequency and its harmonics. In a practical implementation the spike would be broadened by the finite length of the data. To display the small amplitude features it is useful to plot the logarithm of the power. For Z(t) the power spectrum is:

Although there is a relatively high intensity around a frequency of 8 (which corresponds in the time series to the frequency of the orbit circulating either fixed point) and its harmonics, the peak is quite broad, and there is also a broad band component to the spectrum - often used as another way of detecting the "random" component of chaotic dynamics.

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Last modified Tuesday, October 28, 1997
Michael Cross