1 Introduction

A goal of observing solar phenomena is to understand how solar activity occurs. The objective of this study is to compare the 10.7-cm radio flux values [F 10.7] with the International Sunspot Numbers [R i] for solar-activity Cycles 19, 20, and 21.

The Sun’s magnetic field appears concentrated in flux tubes or ropes that appear on the surface of the photosphere as sunspots, pores, plages, and surface network. The magnetic fields that result from the solar dynamo include all solar motions (Ruzmakin 2001; Rogachevskii and Kleeorin 2007). R i and F 10.7 are proxies for phenomena associated with the solar cycle (Eddy 1977).

The International Sunspot Number [R i] is the weighted sum of the number of spots and spot groups on the solar disk at one time:

$$ R_{\mathrm{i}} = k (10g + f), $$
(1)

where f is the total number of spots on the visible disk irrespective of size, g is the number of spot groups, k is a coefficient depending on the observer’s method of counting and subdividing them into groups, the size of the telescope, the magnification, and seeing conditions. The index R i is not the number of spots but a relative number including groups. The sunspot number usually reaches a maximum three or four years after the start of a solar-activity cycle and declines for seven or eight years. Thus the average activity cycle lasts about eleven years. Sunspots are a convenient index for most activity related to the Sun’s magnetic fields (Eddy 1977).

Chaos is described and a brief review of articles utilizing chaos and fractals are presented. Another measure of solar activity is the 10.7-cm radio flux values [F 10.7]. The flux results from thermal ionization at the photosphere–chromosphere interface and from magnetic resonance above sunspots and plages (Tapping and DeTracy 1990). The origin of the 10.7-cm radio flux is discussed and a short review of recent work discussing its measurement and its relation to the International Sunspot Number and other proxies for solar-activity cycles. One would expect R i and F 10.7 to be correlated.

Synoptic radio observations of the Sun were started in Canada in November 1946 and became an observing program of F 10.7 in 1947 (Covington 1948). The values are reported in solar flux units [sfu=10−22 W (m2 Hz)−1]. F 10.7 is used as a proxy for R i, sunspot area, E-Layer ionization index, Lyman β, Mg ii, EU, EUV, and total irradiance (Schmahl and Kundu 1998).

Section 2 presents a brief discussion of fractals and chaos applied to R i and F 10.7.

Section 3 presents the origin of the 10.7-cm radio flux values and includes a brief review of articles that discuss measurement of the radio flux and its relation to the International Sunspot Numbers and other proxies for solar activity. The relation between the proxies is discussed including chaos. Differences, time scales, and fractals are discussed. The use of F 10.7 in prediction models for activity-cycle properties is examined. Section 4 presents a short history of solar-activity cycles as measured by the sunspot index. Section 5 considers the length of the data sets necessary to analyze the data from a statistical or a nonlinear point of view. Section 6 applies nonlinear analysis to F 10.7 for Cycles 19, 20, and 21. F 10.7 for Cycle 19 appears to be stochastic. F 10.7 for Cycles 20 and 21 appear to be chaotic. Section 7 contains a discussion of the results including a day-by-day comparison of F 10.7 with R i. Section 8 presents the conclusions.

2 Fractals and Chaos

A fractal line is not differentiable and is scale independent. A magnified segment resembles the large-scale appearance. The length of a fractal line does not approach a finite limit as the resolution increases but it increases without limit. A familiar example of a complex line is a coastline (Feder 1989). To measure the length [L] of a complex line segment such as a coastline, select a measure of length [δ] and measure along the line segment and obtain the number [N(δ)] of measures [δ] to cover the line. The length [L] is the number of measures times the length of the measure, so L(δ)=N(δ)×δ. As the measure δ is decreased, the number of steps [N] required to cover the coast line increases. The measured length increases as δ is reduced. Plotting the length as a function of δ yields a straight line approximated by

$$ L(\delta) = a \delta^{1-D}. $$
(2)

For an ordinary line, a=L N and D=1. However for the coast line D≈1.52, thus the coast line is a fractal with fractal dimension D. A fractal curve (coast line) is a shape made up of parts similar to the whole. A fractal appears the same regardless of scale.

The set of points that make up a line in ordinary Euclidean space has a topological dimension D T=1. The Euclidean dimension of space is E=3. The set of points that form a surface in E=3 has a topological dimension D=2 and D T=2. For a sphere, D=3 and D T=3. Analytically in the limit of small δ

$$ N(\delta) \approx 1/\delta^{D}. $$
(3)

The fractal dimension of a coast line can be found from the slope of lnN(δ) plotted as a function of lnδ. The fractal dimension of a coast line is 1.52. A fractal looks the same, whatever the scale; it is self-similar.

Chaos is an irregular deterministic motion characterized by a continuous, broad-band Fourier power spectrum in a three or higher dimension nonlinear system. Chaos occurs between regular behavior and stochastic behavior. An attractor is a point or set of points in phase space towards which a system evolves over time. Systems with low dissipation could take a very long time. An attractor can have a regular dimension – a regular attractor, or a fractal dimension – a strange attractor.

The stability of motions on an attractor is determined by the Lyapunov exponents. The stability can be determined from the Jacobian matrix of the dynamics. The eigenvalues of the matrix are determined (Oseledec 1968). The eigenvalues are \(\mathrm{e}^{\lambda_{1}} , \mathrm{e}^{\lambda_{2}} , \ldots, \mathrm{e}^{\lambda_{a}}\), where λ 1≥λ 2≥⋯≥λ d are the global Lyapunov exponents.

Measurements such as daily values of the sunspot number or the 10.7-cm radio flux result in a time sequence sampled at intervals of τ initiated at t 0.

$$ s(t) = s(t_{0} + n \tau) = s(n). $$
(4)

Chaos has structure in phase space. Time-lagged variables are the coordinates for a phase space in d E dimensions,

(5)

where T is time lag and d E is embedding dimension. The global embedding dimension is the lowest dimension that unfolds the attractor (the set of points visited by a signal trajectory) so that there are no overlaps from projection of the attractor to a lower dimension. The embedding theorem (Takens 1980) is: If the dimension of the attractor defined by the orbits is d A the attractor will unfold in an integer-dimensional space of dimension d E, where d E≥2d A.

The two invariants of the motion are the fractal dimension of the attractor and the Lyapunov exponents. The correlation dimension [D 2] is a fractal dimension associated with an inhomogeneous attractor that quantifies self-similarity. The slope of the correlation function is D 2 (Grassberger and Procaccia 1983). Although the correlation dimension [D 2] applies to the attractor, the existence of a scaling region does not necessarily indicate that the data are chaotic. Stochastic data may also have a scaling region.

3 Review

The topics reviewed in this section are: the origin of the 10.7-cm radio flux; solar activity; the relations between proxies for the activity cycles, differences, time scales, and fractals.

3.1 The 10.7-cm Radio Flux Values F 10.7

The integrated emission of centimeter wavelengths from the whole solar disk is separated into three components based on characteristic time scales: transient events such as flares with durations less than an hour; slow variation over hours to years; a continuous minimum level, the quiet-Sun level. The second component with time scales of hours to years is called the S-component (Tapping and DeTracy 1990).

The S-component at 10.7-cm wavelength has been measured since 1947 (Covington 1947). F 10.7 is a measure of solar activity and is a proxy for other measures such as: the sunspot index (Covington 1948); the full-disk flux in Ca ii and Mg ii (Donnelly 1987); the integrated fluxes in ultraviolet (UV) and in extreme ultraviolet (EUV) (Schmahl and Kundu 1998). When the full-disk X-ray flux activity is high, F 10.7 and the X-ray flux are correlated (Tapping and DeTracy 1990).

The intensities of the Ca ii and Mg ii spectral lines are functions of chromospheric density and temperature. X-rays are produced in the corona. The S-component is integrated emission from all sources on the solar disk with contributions from thermal free–free and gyroresonant processes, and some non-thermal emissions. The relative magnitude of these mechanisms is a function of wavelength and time (Tapping and DeTracy 1990). The spatial distribution of the two thermal mechanisms are different. The gyroresonant emission is mainly in the vicinity of sunspots where magnetic fields are strong. Free–free emission is more widely distributed.

The correlation of F 10.7 and R i implies a connection with strong magnetic fields. The correlation of F 10.7 with other phenomena (Ca ii, Mg ii, UV, EUV, and X-rays) implies a relation with widely distributed phenomena. Gyroresonance requires strong magnetic-field strengths, which occur in the vicinity of sunspots. Free–free emissions take place over the entire solar disk – the quiet Sun level (64 sfu) – with enhanced emission produced where the densities in the chromosphere and corona are above the usual values. A value of F 10.7 is apparently an indication of the density in the low corona.

Tapping and DeTracy (1990) construct a two component model: an isothermal chromosphere at T=104 K and a homogeneous corona at T=106 K. Variations of the thickness of the chromosphere, transition region, and mixed zone cause only small changes in the S-component spectrum of the model. There is a strong dependence on plasma density at the base of the corona. Dependence on a few quantities and its origin in the low corona makes it useful as an index of solar activity.

Total-irradiance variation may affect the Earth’s climate. To understand the role of irradiance variation requires a long record. Proxies and/or models are used to generate long records. Most models are based on the assumption of two irradiance variations which may vary with time: the cyclic component that follows R i and F 10.7, and a secular component, the quiet Sun, with variations on time scales much longer than a solar-activity cycle. Extending such models back in time to the Maunder minimum allows one to estimate the secular component at that time. Extrapolations in time assume that the relationship between indices used in the models do not change. Short-term irradiance variation correlates with bright faculae and magnetic networks (Tapping et al. 2007).

The input to the models is R i and F 10.7. There are two approaches to modeling irradiance. One approach assumes that irradiance is a photospheric mechanism (Foukal and Lean 1990). The other approach uses more global variation due to small temperature changes in the photospheric temperature outside active regions. This approach may involve change in the photospheric radius (Ulrich and Bertello 1995).

Rotation of the Sun’s atmosphere is affected by the magnetic structure in the convection zone. Sunspots have been used to determine rotation rates. Changes in the size of the convection-zone radius may be due to changes in the turbulence regime in the convection zone (Greenkorn 2009). F 10.7 can also be used to determine rotation rates. Sunspot measurements provide differential-rotation rates over long periods. Data from the analysis of the F 10.7 flux measurements are frequency distributed with respect to time. There are regions on the Sun with rigid and differential-rotation properties in both the photosphere and the corona (Mouradian, Bocchia, and Botton 2002).

3.2 Solar Activity

F 10.7 is an indicator of radiative variability. The F 10.7 flux is the basis for estimating radiative output when no absolute measurements are available (White and Rottman 2000). Radio observations are used to study the solar atmosphere. They can be used to probe the chromosphere up to the middle of the corona. Radio observations provide information about the magnetic field. Where the magnetic fields are strong, strong emissions can be produced at low harmonics of the electron gyrofrequency. This thermal-electron gyroresonance is bright, narrowband, and highly polarized. This emission is associated with the strong magnetic fields over sunspots (Alissandrakis 1997). In the chromosphere and corona above active regions, the magnetic fields are too weak for gyroresonance. Emission is produced by thermal free–free processes. Polarization is introduced by propagation through the plasma giving additional information about the magnetic field. The beginning of a new activity cycle is the transition from a quiet Sun to an active Sun with a complex magnetic atmosphere. Active longitudes contain sunspot nests. Solar minimum is a time mark in statistical schemes for predicting solar activity. R i,F 10.7, and the Mg ii index follow the increase in the observed magnetic field (DeToma, White, and Harvey 2000).

Solar-activity indices show variation from a few days to several years. The most prominent short-term variation is the 27-day periodicity due to solar rotation. Sunspots and magnetic features rotate faster than the solar plasma, probably due to magnetic linkage to faster rotating sub-surface layers (Howard 1984) or due to buoyancy and drag coupled with Coriolis force (D’Silva and Howard 1994). Bouwer, Pap, and Donnelly (1990) report that magnetic and photospheric rotation is at a period of 25 days, Mg ii and F 10.7 at 27 days, and coronal X-rays at 28 days. Spectral analysis of the time series of 10.7-cm solar radio emissions [F 10.7] and R i show periodicities in the range of 11 to 70 days, which could be due to a combination of active-region durations, the number of complexes or active longitudes, and solar rotation. Group sunspot numbers correlate with the 10.7-cm flux but not as well as with the sunspot index (Kane, Vats, and Sawant 2001). This higher correlation with groups may be due to a significant part of the F 10.7 originating over an active region beyond a sunspot. A group of sunspots is more likely to be associated with a large area of plage over which free–free emission is produced.

3.3 Relation of F 10.7 with Other Proxies

F 10.7 is a single-wavelength measurement of the S-component made close to the spectral peak. Solar soft X-rays and F 10.7 are highly correlated, probably because they both are produced in the solar corona, where the electron temperature exceeds a million Kelvin. The solar soft X-rays and the slowly varying component of the 10.7-cm radio flux are emitted from the solar corona where T≥106 K and they are correlated. F 10.7 is used in atmospheric and ionospheric models as an indicator of solar activity of the UV and EUV radiations (Donnelly 1982; Chatterjee and Das 1995). A statistical analysis of the relation between F 10.7 and R i on a daily basis showed the correlation between them is very high (Xanthakis and Poulakos 1985). Empirical relationships for the calculation of monthly and annual indices of the radio flux and sunspot numbers are available (Kononovich, Shefov, and Khramova 2002). There is also a correlation of the radio flux with plage area (Oster 1990). Despite the strong correlation, the two indices behave differently at solar-activity-cycle maximum. This could be due to a change in the way that magnetic activity is manifested over a solar cycle (Wilson, Rabin, and Moore 1987), such as differences in the ratio of spot area to plage area. Change in the area ratio would change the relative contribution to F 10.7 by gyroresonance and free–free emissions, such as changes in the relative contributions of gyroresonance and free–free emission over the solar cycle.

Global indices of solar activity in terms of nonlinear dynamics show that solar periodicity is a complex dynamic system that may be stochastic at times and chaotic at times. Nonlinear interaction occurs but not between all indices (Salakhutdinova 1999). Rigozo et al. (2001) used a reconstructed sunspot-number series to infer past F 10.7, solar-wind velocity, and the southward component of the interplanetary magnetic field.

3.4 Differences; Time Scales; and Fractals

In any given active region, the sunspot area usually peaks before the plage area and plage is still present after all sunspots have disappeared. Thus the region’s contribution to sunspot related indices, e.g. sunspot number, peaks before the contributions to plage-related indices, e.g. UV flux and Ca ii K index. The F 10.7 solar radio-flux index is more complex because it contains both spot and plage-related contributions. Bright cores of thermal gyroresonance lie over sunspots, and a diffuse cloud of free–free emission overlies the plage. The brightness temperature of the thermal gyroresonance is higher, but the free–free emission comes from a larger area. The result is that the variation with time of the contribution to the F 10.7 index can be multi-peaked and can be complicated. The situation may become more complicated by additional magnetic-flux emergence (Donnelly et al. 1983). This complexity affects the correlation of F 10.7 with the UV flux and of heating and photodissociation of O2 in the stratosphere and mesosphere. Although the radio flux and sunspot index are well correlated, the temporal variation of the radio flux and sunspot index and Ca K plage data are not as well correlated. The UV flux has a 13-day temporal structure. The radio flux and sunspot index do not show the 13-day temporal structure. They do have a broader dependence on the solar central meridian distance (CMD). The CMD-dependence of cm-wavelength emission of active regions differs from the effective-area foreshortening of plages because coronal radio emissions are moderately strong from active regions at the solar limb. Another type of difference in the temporal variation occurs during the evolution of major active regions that dominate the full disk. Simple regression estimates of UV flux based on radio-flux values or sunspot numbers fail to fit the temporal variation of the UV flux. Plages are usually lasting longer than sunspots associated with the same active regions. R i peaks earlier than UV because major active regions of sunspot activity peak earlier than plage activity. Part of the radio-flux emissions are from small intense cores in regions of strong coronal magnetic fields and gyroresonant emissions from active regions tend to peak earlier and decay more rapidly than associated UV plage emissions. Radio emissions from solar active regions are complicated. The radio flux peaks and decays more rapidly than plage emissions and behaves more like sunspot evolution. Although there are some similarities among the time scales of UV fluxes, radio-flux emissions, and sunspot numbers on rotation and evolution, there are differences. Strong 13-day modulation lasting months occurs. There are differences due to CMD dependence. Simple linear regression estimates of UV flux based on R i and F 10.7 fail to fit the temporal variation of the UV flux (Donnelly et al. 1983).

The green coronal brightness indices [CI: Fe xiv, λ=530.3 nm] in the minima of Cycles 19 – 20, 20 – 21, and 21 – 22 were compared with R i and F 10.7 for 1964 – 1987. The patterns of the three are similar. The minimum between Cycles 21 – 22 is different from that between the two previous minima. The most significant changes were in CI, which may be a better proxy to study the Sun (Rušin and Rybansky 1990).

Fractals were used to examine R i (1854 – 1996), total sunspot area (1874 – 1983), F 10.7 (1964 – 1996), and Stanford mean magnetic field (1975 – 1996) (Salakhutdinova 1998). Similar values of the fractal dimensions of these indices on different time scales were determined. For sunspot numbers, total sunspot area, and radio flux there are three time scales: two days – two months, two months – two years, and > two years with values of fractal dimensions of 1.5, 2, 1.2 – 1.6 respectively.

Fractals are used to examine R i,F 10.7, Ly α, Fe xiv, and total solar irradiance (Watari 1995). The fractal dimension approaches two if the time series is random and becomes one for a regular time series. The length of the curve for a time interval k is the average value over sets of L m (k). If L(k)∼k −D, the curve is fractal with dimension D. The plot of L(k) may have one or more straight-line sections. The straight lines are joined at a point where the slope changes: the bending point. The fractal dimension of the 10.7-cm radio flux in Cycle 21 for periods of ≈ seven days or less was 1.28; for periods longer than 272 days was 1.3, and for the interval between them was 1.86. Fractal dimensions for the other indices were similar. For both short and long time scales, the bending points for the radio flux and the sunspot numbers are similar. Two fractal dimensions and one bending point can be found for the daily indices and three fractal dimensions and two bending points can be found for a cycle. The variations of fractal dimensions and bending points for CI are different from the radio flux and sunspot numbers. Fractal dimensions for the sunspot numbers are ≈ 1.5 for both short and long periods. Solar activity is irregular for the time scales between several days and several months. The fractal dimensions for long time scales may reflect a long-term variation due to large solar-scale features. Yearly values of D and bending points of 〈L(k)〉 do not change with solar cycle.

Watari (1996) calculated the fractal dimensions of the solar radio fluxes for 1976 – 1990 for nine values, ranging from 245 to 15 400 MHz. The fractal values determined were 1.2 – 2.0 for ≤ ten days, one – ten months, and ≥ ten months. The quiet-Sun emission at different frequencies contains information about densities and temperature in different levels of the solar atmosphere. The strength of the solar radio emission, the S-component, varies with the 11-year solar cycle. The plage-associated emission dominates around 1 – 2 GHz and sunspot-associated emission dominates around 5 GHz. Both the chromosphere and corona contribute to the quiet-Sun emission at decimeter wavelengths. The fractal dimensions of the sunspot number and solar radio flux around 3 GHz show similar behavior, confirming the existence of a close relation between R i and the radio flux at 2.8 GHz (10.7-cm flux). The quiet-Sun emission at different frequencies contains information about densities and temperature in different layers of the solar atmosphere.

4 Solar Activity Cycles

Records of sunspot numbers and aurorae can be used to estimate sunspot-number maxima from AD 300. Sunspot number maxima from 649 AD to AD 2000 were estimated, one in multiples of 11.1 years and another of ≈ 78 years. The length of a sunspot-number cycle is longer in weak-aurora periods (Shove 1955). Tables of sunspot numbers have been estimated for 1610 to 1960 (Waldmeier 1961). Sunspot numbers may arise from a stochastic process (Siscoe 1976) or from a deterministic chaotic process (Feynman and Gabriel 1990). Chaotic behavior results from competition between linear instability and nonlinear suppression of cycles (Kremliovski, 1994, 1995). Describing activity cycles with sunspot numbers or group sunspot number gives similar results (Hoyt and Schatten, 1988a, 1988b).

Short periodicities in sunspot numbers of 27, 30, 50, and 150 days have been identified (Torrence and Campo 1998). Longer-term periodicities of 50 – 80, 90 – 120, and 120 – 260 days may have existed for 10 000 years (Ogurtsov et al. 2002). Hemispheric solar activity is asymmetric with each cycle but approximate 11-year cycles (Li, Yun, and Gu 2001). There are other proxies for solar activity including sunspot areas, the 10.7-cm radio flux, the solar wind, the cosmogenic isotopes 10Be and 14C, the geomagnetic aa index, solar irradiance, the Mg ii index, magnetic flux, aurora, and the flare index (Eddy 1977).

5 Length of Data Sets

For statistical treatment of the stochastic regime, the length of the data set depends on the persistence and conservation of the time series of the data. Chapman and Bartels (1940) derived the equivalent number of repetitions [ε(h)] and the effective number of random coordinates, [h/ε(h)], where m is the standard deviation of the averages of the numbers:

$$ \varepsilon(h) = \bigl[m(h)/(m/\sqrt{h}) \bigr]^{2} = hm^{2}(h)/m^{2} $$
(6)

and showed that a data set with the length of an activity cycle, about 11 years, has only 20 independent values. This means that to use random statistical methods to analyze data, the set must be very long, much longer than an activity cycle.

The correlation dimension [D 2] is a fractal dimension

$$ D_{2} = \mathrm{d} \log\bigl[C(r)\bigr]/ \mathrm{d} \log r, $$
(7)

where

$$ C(q,r) = r^{(q-1)/Dq} . $$
(8)

For q=2, the number of data points [N] needed (Eckman and Ruelle 1992) is at least

$$ 2.3 \ln N = D_{2} /2. $$
(9)

According to Equation (9) the number of data points to cover the attractor and avoid sample-size problems is about 1000 and a single activity cycle (11 years or 4096 days) has enough points to estimate D 2 and determine whether a cycle is chaotic.

The local Lyapunov exponents are functions of x and L, where L is the total length from each calculation starting point on an attractor where evaluation of the Lyapunov exponents begins. An average over all of these starts yields the average Lyapunov exponents. L=1024 is usually enough (Abarbanel 1996). Possible effects of sample size are avoided by using a long time series. There are no precise criteria for determining the size of the data set for calculating the exponents for a particular dynamic system. As a general rule, the number of data points increases with the dimension of the attractor (Brown, Bryant, and Abarbanel 1991):

$$ \log N_{\mathrm{D}} \approx d_{\mathrm{A}}. $$
(10)

Equation (10) gives an estimate of the number of data points large enough to cover the attractor and avoid sample-size problems. For d A=D 2∼D L=3, where D L is the Lyapunov dimension, and N D=1000, so the number of points in a single cycle is probably high enough to determine the Lyapunov exponents.

6 Nonlinear Analysis of the 10.7-cm Radio Flux for Cycles 19, 20, and 21

This section describes the nonlinear analysis applied to the 10.7-cm radio flux values for Cycles 19, 20, and 21. Appendix B of Greenkorn (2009) describes in detail the calculation algorithms of Abarbanel (1996) and Abarbanel, Frison, and Tsmring (1998). The algorithms are used to calculate the optimum time lag, embedding dimension, local dynamical dimension, the correlation sum and its derivative, and the Lyapunov dimensions.

To convert a time sequence such as Equation (4) to phase space, a time delay must be selected. A prescription for selecting the time delay is the first minimum of the average mutual information versus time lag T,

$$ I_{AB} (T) = \sum_{ab} P_{AB}(a,b) \log_{2}\bigl[P_{{A}}(a)P_{B}(b)/P_{AB}(a,b)\bigr], $$
(11)

where a is at time k and b is at k+T,P AB is the joint probability of A and B,P A and P B are the probabilities of A and B. The lowest dimension that unfolds the attractor so that there are no overlaps from projection of the attractor to a lower dimension space is the global embedding dimension.

The stability of motions on the attractor is determined by the Lyapunov exponents defined in Section 2. Dissipation in the dynamical system requires

$$ \sum_{a=1}^{d} \lambda_{a} < 0 , $$
(12)

so volumes in phase space shrink with time. If one of the λ a is greater than zero, the dynamical system may be chaotic. If one of the λ a is zero, the system can be described by a set of ordinary differential equations. Evaluating the average Lyapunov exponents forward and backward gives d L that have the same value but with a sign change. These are the correct exponents. There may be d E−d L that do not change sign. If the sum of the exponents is positive the system is probably stochastic.

The Lyapunov dimension [D L] (Kaplan and Yorke 1979) can be associated with a volume in phase space that does not change. D L is an estimate of the fractal dimension [d A] of the system:

$$ D_{\mathrm{L}} = K + \sum_{a = 1}^{K} \lambda_{a}/\lambda_{K+1}. $$
(13)

7 Discussion

The optimum time delay [T OP], embedding dimension [d E], local dimension [d L], Lyapunov dimension [D L], and the sum of the Lyapunov exponents were calculated for the daily values of the 10.7-cm radio flux for Cycles 19, 20, and 21 using the flux values from the radio flux data in “Solar Variability Affecting Earth” National Oceanic and Atmospheric Administration (NOAA) CD Rom NGDC-05/1. The number of flux values and sunspot numbers used for Cycle 19 is 3872, for Cycle 20 is 4201, and for Cycle 21 is 3761. The Lyapunov exponents are determined by calculating the eigenvalues of the Jacobian matrix of the dynamics. The Lyapunov dimensions are calculated using Equation (13) and the spectrum of Lyapunov exponents. The values of D L versus time lag [T] are plotted in Figure 1 for Cycle 20 for embedding dimensions of 7, 6, and 5. The values of D L versus time lag are plotted in Figure 2 for Cycle 21 for embedding dimensions of 6 and 5. The values of D L versus time lag T are plotted in Figure 3 for the combined values in Cycles 20 and 21 for embedding dimensions of 7 and 6.

Figure 1
figure 1

10.7-cm radio flux values – Lyapunov D L versus lag time T for Cycle 20 with embedding dimension of 7 (a), 6 (b), and 5 (c).

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Figure 2
figure 2

10.7-cm radio flux values – Lyapunov D L versus lag time T for Cycle 21 with embedding dimension of 6 (a) and 5 (b).

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Figure 3
figure 3

10.7-cm radio flux values – Lyapunov D L versus lag time T for Cycles 20+21 with embedding dimension of 7 (a) and 6 (b).

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In all of these plots, D L is leveling off at a time lag of four. This value of D L is the fractal dimension of the attractor.

The values of D L versus time lag T for Cycle 20 were determined for embedding dimension 7 and local dimensions 7, 6, 5, 4, and 3; for embedding dimension 6 and local dimensions 6, 5, 4, and 3; for embedding dimension 5 and local dimensions 5, 4, and 3 for the radio flux values and sunspot numbers. Figure 4 is an example of the comparison of the radio flux and the sunspot number for embedding dimensions 7, 6, and 5 and local dimension 5F (radio flux) and 5R (sunspot number) for Cycle 20. Figure 5 is an example of the comparison of the radio flux and the sunspot number for embedding dimensions 7, 6, and 5 and local dimension 5F (radio flux) and 5R (sunspot number) for Cycle 21. The embedding dimension at the optimum time delay was determined by finding the dimension for which the number of false nearest neighbors approaches zero. The local dimension is determined by calculating where the percentage of bad predictions becomes independent of the dimension and number of neighbors.

Figure 4
figure 4

Comparison of 10.7-cm radio flux values [F 10.7] and International Sunspot Numbers [R i] for Cycle 20 with embedding dimension of 7 (a), 6 (b), and 5 (c) and local dimension 5F and 5R.

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Figure 5
figure 5

Comparison of 10.7-cm radio flux values [F 10.7] and International Sunspot Numbers [R i] for Cycle 21 with embedding dimension of 7 (a), 6 (b), and 5 (c) and local dimension of 5F and 5R.

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A day-by-day comparison of the sunspot numbers and the radio-flux values was accomplished as follows. The radio-flux values and the sunspot numbers were each made dimensionless by dividing each value in each cycle by the maximum value in that cycle obtaining R i/R imax and F 10.7/F 10.7max. (The maximum values for R i and F 10.7 occurred on the same day for Cycle 20 and one day apart for Cycle 21.) The difference between the dimensionless values of the sunspot numbers and the radio-flux numbers was calculated for each of the cycles. The differences are plotted versus cycle day in Figure 6 for Cycle 19, Figure 7 for Cycle 20, and Figure 8 for Cycle 21 (if the differences were all zero), there would be perfect correspondence.

Figure 6
figure 6

Difference between dimensionless [R i] and dimensionless [F 10.7] for Cycle 19, starting on 20 April 1954.

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Figure 7
figure 7

Difference between dimensionless [R i] and dimensionless [F 10.7] for Cycle 20, starting on 1 January 1965.

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Figure 8
figure 8

Difference between dimensionless [R i] and dimensionless [F 10.7] for Cycle 21, starting on 3 July 1976.

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8 Conclusions

For Cycle 19, the sum of the Lyapunov exponents is positive indicating that the flux values are stochastic. For Cycle 20 there is a positive Lyapunov exponent, the sum of the Lyapunov exponents is negative, and the Lyapunov dimensions are fractal for embedding dimensions 7, 6, and 5 indicating that the flux values are chaotic. For Cycle 21 there is a positive exponent, the sum of the Lyapunov exponents is negative, and the Lyapunov dimensions are fractal for embedding dimensions 6 and 5 indicating that the flux values are chaotic. The embedding dimension at the optimum time delay was determined by finding the dimension for which the number of false nearest neighbors approaches zero. The local dimension is determined by calculating where the percentage of bad predictions becomes independent of the dimension and number of neighbors.

Generally the plots of D L versus T for all combinations are similar to the plots shown in Figures 4 and 5. The two sets (5F and 5R) are parallel with very close values at the maximum time lag. Usually the value for the sunspot number is higher than the radio flux but within 0.2 of the radio flux value. The plots show that the radio flux-values and sunspot numbers are in agreement. The differences in each of Figures 6, 7, and 8 show a trend line with a negative slope, which indicates a more complex relation between the two dimensionless values than if the slope were zero.

The radio-flux values indicate that there is a transition from stochastic to chaotic behavior between Cycles 19 and 20. The sunspot numbers indicated the same transition (Greenkorn 2009). The radio-flux values seem more consistent showing chaotic behavior than the sunspot numbers. The radio-flux values may be a better indicator of magnetic changes than the sunspot numbers and a more consistent proxy for solar activity. The change in the radio-flux values between Cycles 19 and 20 implies, as did the sunspot numbers, a change in the scale of turbulence in the convection region. The differences in each of Figures 6, 7, and 8 shows a trend line with a negative slope, which indicates a more complex relation between the two dimensionless values than if the slope were zero.