# Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum--An Inverse Problem
## 5 Coupled Periodicities Using a Bispectrum  
Analyzing the bispectral contents of the sunspot series described in Section 2, we created three bispectral representations, which are demonstrated in Figures 9-11, and are actually one subset of the next representation. This was necessary in order to be able to follow the bispectral content as we zoomed out of larger areas of frequencies. This bispectral representation was preprocessed with the point operation \(log_{e}(1+x)\) in order to suppress very large components and simultaneously enhance weak peaks. Figure 9: Bispectrum of the sunspot series containing the first 200 harmonics. On both axes, the 200 th harmonic corresponds to 32,000/200 = 160 days. We obtained symmetry with respect to the diagonal line with an origin at (1,1). All frequency coupling edges that we are interested in lie in the orthogonal red triangle. The four elongated ellipses contain specially coupled groups of periodicities, as explained in the text. Figure 11: Bispectrum of the sunspot series containing the first 1300 harmonics. On both axes, the 1200 th harmonic corresponds to 32,000/1200 = 26.7 days (rotation of the Sun at its equator). We achieve symmetry with respect to the diagonal line with its origin at (1,1). Figure 10: Bispectrum of the sunspot series containing the first 400 harmonics. On both axes, the 400 th harmonic corresponds to 32,000/400 = 80 days. We achieve symmetry with respect to the diagonal line with its origin at (1,1). Accordingly, the bispectral representation in Figure 9 provides frequencies that correspond to periodicities up to those of Earth and Venus (frequency lines 88 and 143, respectively). The peaks in this figure correspond to a large variety of frequency couplings. We concentrated our attention on four different sets of them, which presented particular groupings. Findings:  
1. Peaks that appear in the vertical, green ellipse in Figure 9. These peaks correspond to couplings of the frequency component 140 with most of the lower frequencies and some of the higher ones. Frequency component 140 corresponds to Venus' period and simultaneously to the synodic periods of Venus with large planets (Jupiter, Saturn, Uranus, and Neptune). This coupling is very strong and probably imposed by the Sun's interior plasma circulation. Obviously, the strong signature of the periodic movements of Venus and the corresponding couplings on the bispectral representation are strictly related to the strong signature of the same periods on the spectrum of the sunspot data as they are depicted in Figure 7 (peaks with numbers in the range of 20-24). 2. Peaks that appear in the vertical yellow ellipse in Figure 9 at position 42. These peaks correspond to couplings of the frequency component 42 with most of the frequencies present in the spectrum. Frequency component 42 corresponds to Mars' synodic periods with Earth and Jupiter. This coupling is very strong and probably imposed by the Sun's interior plasma circulation. Obviously, the strong signature of the synodic movements of Mars and the corresponding couplings on the bispectral representation are strictly related to the strong signature of the same periods on the spectra of the sunspot data as they are depicted in Figure 7 (peaks with numbers 7 and 8). 3. Peaks that appear in the vertical yellow ellipse in Figure 9 at position 8. These peaks mainly correspond to the coupling of the eleven-year cycle with all frequencies present in the sunspot time-series spectrum. Simultaneously, these couplings contribute the periodic movement of Jupiter as well as the synodic periods of Jupiter with Neptune and Uranus. 4. It is worth mentioning that the bispectrum from 1 (dc) to the 7 th frequency component is very rich since it represents the couplings of all periodic movements of the spectra with the periods and synodic periods of the large and distant planets (Jupiter, Saturn, Uranus, and Neptune). 5. Peaks that appear in the diagonal red ellipse in Figure 9. These bispectral peaks can be described by the equation \(f_{1}+f_{2}=f_{140}\).