Testing the Deterministic Component of the Time Series of Wolf Numbers by Methods of Algorithmic Complexity

Abstract

This article discusses the testing of the complexity of the instrumental series of Wolf numbers. The work is initiated by the hypothesis of the existence of a low-dimensional dynamo as a model of the Sun’s magnetic activity. This mechanism produces the observable, in the Takens sense, as a broadband chaotic signal with a dominant 11-year mode (Frick et al., 2022). The time series of Wolf numbers is claimed to be this signal. In this article, we consider two problems. In the first, we describe a method for obtaining the average cycle for the dominant 11-year mode. It is based on the Fisher–Rao metric and the quantum mechanical analog of “probability amplitudes” for cycles. In the second problem, we investigate the algorithmic complexity of the instrumental series of Wolf numbers (SSN2) compared with surrogate data obtained by fractal mixing of this series. The mixing “whitens” the 11-year cycle but retains tuples of 2–3 monthly mean counts. Complexity was estimated as permutation entropy (Bandt et al., 2002). It was hypothesized that if the dominant mode was chaotic in nature, the complexity of the source and surrogate series would be close. Our results do not contradict the hypothesis of a chaotic signal with a single prevalent mode as a time series model of Wolf numbers.