Multiscale entropy analysis and long-range correlation of different sunspot cycles

Document Type : Research Article

Author

Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), University of Maragheh, Maragheh, Iran.

Abstract

The sunspot system represents a quintessential example of astrophysical complexity, where nonlinear interactions within the solar dynamo generate emergent patterns across multiple spatiotemporal scales. Our investigation of ten solar cycles (15-24) through the lens of multiscale entropy (MSE) analysis reveals profound insights into this complex system. We collected sunspot data (the number of daily sunspots and daily standard deviation of sunspot counting) from the Royal Observatory of Belgium's SILSO database, and decomposed the full time series into individual cycles. Then, we applied MSE approach to time series of individual cycles of sunspots. MSE analysis has two main steps: 1-coarse-graining the time series, and 2-calculating sample entropy for each coarse-grained series. The MSEs of cycles are computed up to scale factor τ=20 and tolerance r=0.15–parameters carefully chosen to capture how information complexity evolves under progressive coarse-graining procedure. This approach is particularly suited to non-stationary systems where traditional entropy measures fail, as it quantifies how structural regularity changes when observed at different temporal resolutions.
For studying the long-range correlated behavior of solar cycles, we employed rescaled range (R/S) analysis. It calculates the Hurst exponent (H), which characterizes the system's fractal geometry and persistence (0.5<H<1) or anti-persistence (H<0.5). Using this approach, obtained Hurst exponents ranged in 0.81-0.86 across all different cycles, providing robust mathematical evidence of persistent long-range memory. This persistent behavior which is valid for Hurst exponents ranged in (0.5, 1) signifies fractal organization where correlations follow a power-law decay. Such scaling behavior implies that sunspot dynamics exhibit 1) statistical self-similarity across observational timescales, and 2) information encoding where subsystem behavior reflects global organization. These characteristics align precisely with self-organized criticality (SOC) – a universal mechanism where driven-dissipative systems spontaneously evolve toward critical states characterized by scale-invariant fluctuations. So, within this framework, it is revealed that the different cycles of solar activity have long-term memory in their time series. Since the sunspots have the magnetic origin, we can say that the solar magnetic field operates near a critical threshold, enabling energy release through avalanches of magnetic reconnection events.
Furthermore, it was discovered that neither mean daily sunspot number nor mean standard deviation of daily sunspot number have no meaningful relation with MSE-derived complexity. Quantifying complexity through the area under MSE curves revealed Cycle 20 as exhibiting peak complexity; while Cycle 24 showed minimal complexity. Using proposed methods, we found that among all cycles of solar activity with long-range correlated behaviors, Cycles 20 and 24 had the maximum and minimum long-range memory in their time series, respectively. The power-law exponential function as  is fitted to ensemble-averaged entropy profiles. The exponential term induces an overall decreasing trend, yet its form creates a dynamic equilibrium between growth and decay up to scale factor τ=20. This functional behavior captures an initial growth phase followed by saturation, potentially indicating system stability. Crucially, it highlights that while complexity evolves differently across scales in various systems, it is universally moderated by a limiting factor.

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References

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Journal of the Earth and Space Physics
Volume 52, Issue 1
online publication date: 30 May 2026
May 2026
Pages 107-120

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