# Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum--An Inverse Problem
## 1 Introduction  
The Sun undergoes cyclical changes, and one well-known cycle is the 11-year cycle, as documented by [PERSON] [1]. This cycle is closely linked to the frequency of sunspots that manifest on the Sun's surface. During the solar maximum, which marks the peak of each cycle, sunspots become more prevalent, while during the solar minimum, at the cycle's end (and the beginning of a new one), sunspots are scarce and can even vanish entirely. The recognition of this cycle dates back to the late 18 th century when the Danish astronomer [PERSON] observed a recurring pattern of sunspot numbers fluctuating over several years. Since then, the Sun has completed 24 full cycles, and the 25 th cycle, our current one, commenced in December 2019, with the predicted maximum expected in July 2025. The 26 th cycle is projected to start in 2030. Although the concept of the solar cycle is straightforward, the underlying mechanisms that propel it are intricate and not fully comprehended. However, at its core lies the Sun's magnetic field. The Sun rotates around its axis, but owing to its lack of a solid surface (as well the convective zone up to the tachocline), distinct regions rotate at varying speeds. Specifically, the equator experiences a faster rotation (25.6 days) compared to the poles (33.5 days). Thisrotational difference results in the Sun's magnetic field becoming twisted, causing it to penetrate the solar surface and give rise to sunspots. The Sun's magnetic field activity correlates with the appearance of sunspots--the more pronounced the magnetic field, the more sunspots manifest. Therefore, the solar cycle essentially coincides with the cycle of the Sun's magnetic field. Approximately every 11 years, the magnetic field experiences a quiet phase, known as the solar minimum. In the middle of this period, the solar maximum occurs [1], which corresponds to the most turbulent phase of the magnetic field. The cycle concludes, initiating a new one with another solar minimum. The generation of the magnetic field itself is attributed to the movement of charged particles (plasma) at the convective part of the Sun. As these particles interact, they create electrical currents, generating magnetic fields. However, the precise causes of these fluctuations in the magnetic field remain somewhat enigmatic. Despite the Sun being 93 million miles (150 million kilometers) away, it exerts a noticeable influence on Earth, and alterations in solar activity can impact our planet. Recent endeavors to model solar total and spectral irradiance values across time spans ranging from days to centuries and beyond are detailed in the work by [PERSON] et al. [2]. Extending these models further into the distant past is crucial for providing inputs to climate simulations. Initial strides have also been made in reconstructing solar total and spectral irradiance values over millennia. [PERSON] et al. [3] conducted a comprehensive exploration of the periodicities of solar activity and radiation, reaching back centuries and millennia. This research incorporates the data from various sources, including the Holocene data of solar irradiance derived from the abundance of the 14C isotope. [PERSON] [4] provides a comprehensive review of the current understanding of solar activity's long-term behavior on a multi-millennial scale, reconstructed using the indirect proxy method. [PERSON] et al. [5] conducted an analysis of daily data from January 1995 to December 2018, exploring the periodic behavior and the relationship between sunspot numbers, cosmic ray intensity, and solar wind speed. [PERSON] et al. [6] presented arithmetic expressions for the quasi-11-year cycle, 110-year century cycle of relative sunspot numbers, and quasi-22-year cycle of solar magnetic field polarity through a spectrum analysis. Furthermore, [PERSON] et al. [7] confirmed the distinguishability of two long-term variations in solar activity, namely the Gleissberg and Suess cycles, at least over the last millennium.