# Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum--An Inverse Problem
## 6 Conclusions  
A detailed investigation of the SILSO sunspot time series was carried out based on two well-established frequency analysis tools, namely the power spectrum and the bispectrum. The material in each section was different; there were different types of processing steps and we reached conclusions that were not strictly related. The first one has to do with the spectral content of the SILSO data, while the second has to do with the bispectral content of the SILSO data and the revealed coupled periodicities.  
The power spectrum was employed in its periodogram version; simultaneously, padding zeros at both sides of the data segments were used so that a high-resolution spectral analysis was performed. The results obtained from the spectral density demonstrate that the planetary periodicities as well as the synodic periodicities are mirrored by the sunspot activities on the Sun's surface. In this fine and detailed spectrum calculation, we obtained all the planetary periodic phenomena present on the surface of the Sun. The main contribution of this work is that the periodic phenomena revealed by the sunspot data are limited within the time period of the 11-year cycle as well as the periodicities of the orbits of the planets, showing the possible effect of the orbits of the planets on the Sun's plasma. This contribution fills a gap that exists in the research in relation to the appearance of planetary periodicities in sunspot time series, since, to date, dominant studies have dealt with much longer periodicities (e.g., millennia or centuries) and were based on the way the solar magnetic field in the heliosphere modulates cosmic rays, influencing the production rate of radionuclides on Earth [4; 5; 6; 7; 8; 9; 10; 11; 12].  
As a general inference, even though there is approximately the same number of previous references ([8; 9; 10; 11; 12; 13; 14; 15]) \"for\" and \"against\" our main claim, we observed the connection between the sunspot periods and characteristic periods of planets. However, additional examinations and analyses are needed to confirm that certain periodicities in the sunspot data are really the consequence of the tidal effects of the planets on solar plasma.  
Coupled periodicities were detected on the sunspot time series by means of a bispectrum. The bispectrum revealed the coupling of two distinct frequencies, \(f_{1}\) and \(f_{2}\), which contributes to a new frequency of the spectrum at position \((f_{1}+f_{2})\), by presenting a peak at coordinates \((f_{1}\), \(f_{2})\) of the bispectrum. Accordingly, Mercury, Venus, and Mars periodicities present couplings with the vast majority of the strong line frequencies of the spectrum (groups in vertical ellipses in Figures 9 and 10). Furthermore, the groups of bispectral spikes lying in the diagonal ellipses provide us with the excellent information that all these spikes come from the spectral equation \(f_{1}+f_{2}=f_{coupled}\). This \(f_{coupled}\) corresponds to frequencies of 140, 340, and 1200. According to these observations:_There must exist a significant quadratic coupling nonlinearity, which applies to almost all periodicities that are found in the sunspot time series in such a way that intensive coupled harmonics appear only for specific summations, namely\(f_{1}+f_{2}=f_{140}\), \(f_{1}+f_{2}=f_{340}andf_{1}+f_{2}=f_{1200}\)._  
Conceptualization, S.T., G.K. and V.A.; methodology, G.K. and V.A.; resources, S.T.; writing--original draft preparation, S.T.; writing--review and editing, G.K. and V.A. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Data availability Statement: Data are available upon request.  
The authors declare no conflicts of interest.