### Article

## Stochastic Description of the High-Frequency Content of Daily Sunspots and Evidence for Regime Changes

The irregularity index λ is applied to the high-frequency content of daily sunspot numbers ISSN. This λ is

a modification of the standard maximal Lyapunov exponent. It is here computed as a function of embedding

dimension m, within four-year time windows centered at the maxima of Schwabe cycles. The λ(m) curves form

separate clusters (pre-1923 and post-1933). This supports a regime transition and narrows its occurrence to cycle

16, preceding the growth of activity leading to the Modern Maximum. The two regimes are reproduced by a simple

autoregressive process AR(1), with the mean of Poisson noise undergoing 11 yr modulation. The autocorrelation

a of the process (linked to sunspot lifetime) is a ≈ 0.8 for 18501923 and ≈0.95 for 19332013. The AR(1) model

suggests that groups of spots appear with a Poisson rate and disappear at a constant rate. We further applied the

irregularity index to the daily sunspot group number series for the northern and southern hemispheres, provided

by the Greenwich Royal Observatory (RGO), in order to study a possible desynchronization. Correlations between

the north and south λ(m) curves vary quite strongly with time and indeed show desynchronization. This may

reflect a slow change in the dimension of an underlying dynamical system. The ISSN and RGO series of group

numbers do not imply an identical mechanism, but both uncover a regime change at a similar time. Computation

of the irregularity index near the maximum of cycle 24 will help in checking whether yet another regime change is

under way.

Radio-observations allow us to reveal the long-lived (2–5 days) intersunspot sources (ISS), whose centers are often located above the neutral line of the magnetic field separating leading and following parts of a whole active region (the first type of ISS (ISS-I)) or above the neutral line separating magnetic polarities into complex sunspots (the second type of ISS (ISS-II)). ISS-I and ISS-II demonstrate gyrocyclotron or gyrosynchrotron spectra, more dynamic pre-flare behavior than ISS-III with bremsstrahlung in the quiet active regions. The qualitative model of “three magnetic fluxes” explaining the origin of accelerated particles in ISS and their long-lasting existence and spectral features is proposed.

This paper studies the first differences *w(t)* of the International Sunspot Numbers daily series, ISSN, over the 1850-2013 time span. The one-day correlations *r1* between *w*(*t*) and *w*(*t*+1) are computed within 4 year sliding windows and found to shift from negative to positive values near the end of Cycle 17. They remain positive during the last Grand Maximum and until ~2009, when they fall to zero. We test an autoregressive process of order 1 (AR(1)) as a model that can reproduce the high frequency component of ISSN: we compute *r1* for this AR(1) process, and find that it is negative. Positive values of *r1* are found only if the process involves positive correlation: this leads us to suggest that the births of successive spots are positively correlated. We also show that the two-day correlation *r2 *of ISSN is, as expected, closer to 0 than *r1*. Finally, we identify two prominent regime changes in ~1915 and ~2009, strengthening previous evidence of major anomalies of solar activity at these dates.

We have recently introduced an irregularity index *λ *for daily sunspot numbers *ISSN*, derived from the well-known Lyapunov exponent, that attempts to reflect irregularities in the chaotic process of solar activity. Like the Lyapunov exponent, the irregularity index is computed from the data for different embedding dimensions *m *(2-32). When *m* = 2, *λ* maxima match *ISSN* maxima of the Schwabe cycle, whereas when *m* = 3, *λ *maxima occur at *ISSN* minima. The patterns of *λ* as a function of time remain similar from *m* = 4 to 16: the dynamics of *λ *change between 1915 and 1935, separating two regimes, one from 1850 to 1915 and the other from 1935 to 2005, in which *λ* retains a similar structure. A sharp peak occurs at the time of the *ISSN* minimum between cycles 23 and 24, possibly a precursor of unusual cycle 24 and maybe a new regime change. *λ* is significantly smaller during the ascending and descending phases of solar cycles. Differences in values of the irregularity index observed for different cycles reflect differences in correlations in sunspot series at a scale much less than the 4-yr sliding window used in computing them; the lifetime of sunspots provides a source of correlation at that time scale. The burst of short-term irregularity evidenced by the strong *l*-peak at the minimum of cycle 23-24 would reflect a decrease in correlation at the time scale of several days rather than a change in the shape of the cycle.

We define, calculate and analyze irregularity indices λISSN of daily series of the International Sunspot Number ISSN as a function of increasing smoothing from *N* = 162 to 648 days. The irregularity indices λ are computed within 4-year sliding windows, with embedding dimensions *m* = 1 and 2. λISSN displays Schwabe cycles with ~5.5-year variations ("half Schwabe variations" HSV). The mean of λISSN undergoes a downward step and the amplitude of its variations strongly decreases around 1930. We observe changes in the ratio *R* of the mean amplitude of λ peaks at solar cycle minima with respect to peaks at solar maxima as a function of date, embedding dimension and, importantly, smoothing parameter *N*. We identify two distinct regimes, called Q1 and Q2, defined mainly by the evolution of *R* as a function of *N*: Q1, with increasing HSV behavior and *R* value as *N* is increased, occurs before 1915–1930; and Q2, with decreasing HSV behavior and *R* value as *N* is increased, occurs after ~1975. We attempt to account for these observations with an autoregressive (order 1) model with Poissonian noise and a mean modulated by two sine waves of periods *T*1 and *T*2 (*T*1 = 11 years, and intermediate *T*2 is tuned to mimic quasi-biennial oscillations QBO). The model can generate both Q1 and Q2 regimes. When *m* = 1, HSV appears in the absence of *T*2 variations. When *m* = 2, Q1 occurs when *T*2 variations are present, whereas Q2 occurs when *T*2 variations are suppressed. We propose that the HSV behavior of the irregularity index of ISSN may be linked to the presence of strong QBO before 1915–1930, a transition and their disappearance around 1975, corresponding to a change in regime of solar activity.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.