To give an overview of the long-term change in periodicity in planetary orbital motion， we performed many fast Fourier transformations (FFTs) along the time axis， and superposed the resulting periodgrams to draw two-dimensional time–frequency maps. The specific approach to drawing these time–frequency maps in this paper is very simple – much simpler than the wavelet analysis or Laskar's (1990， 1993) frequency analysis.
Divide the low-pass filtered orbital data into many fragments of the same length. The length of each data segment should be a multiple of 2 in order to apply the FFT.
Each fragment of the data has a large overlapping part: for example， when the ith data begins from t=ti and ends at t=ti+T， the next data segment ranges from ti+δT≤ti+δT+T， where δT?T. We continue this division until we reach a certain number N by which tn+T reaches the total integration length.
We apply an FFT to each of the data fragments， and obtain n frequency diagrams.
In each frequency diagram obtained above， the strength of periodicity can be replaced by a grey-scale (or colour) chart.
We perform the replacement， and connect all the grey-scale (or colour) charts into one graph for each integration. The horizontal axis of these new graphs should be the time， i.e. the starting times of each fragment of data (ti， where i= 1，…， n). The vertical axis represents the period (or frequency) of the oscillation of orbital elements.
We have adopted an FFT because of its overwhelming speed， since the amount of numerical data to be deposed into frequency ponents is terribly huge (several tens of Gbytes).
A typical example of the time–frequency map created by the above procedures is shown in a grey-scale diagram as Fig. 5， which shows the variation of periodicity in the eccentricity and inclination of Earth in N+2 integration. In Fig. 5， the dark area shows that at the time indicated by the value on the abscissa， the periodicity indicated by the ordinate is stronger than in the lighter area around it. We can recognize from this map that the periodicity of the eccentricity and inclination of Earth only changes slightly over the entire period covered by the N+2 integration. This nearly regular trend is qualitatively the same in other integrations and for other planets， although typical frequencies differ planet by planet and element by element.
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