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\title{Chaos in the Solar Cycle: using data to drive predictions}
\author{Matthew Young }
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
This report presents first steps at applying chaos theory to the solar activity cycle. We provide brief reviews of the history of sunspot records, the mathematical framework of chaos theory, and solar dynamo theory. Finally, we present results of nonlinear analysis of solar motion and propose that further research focus on characterizing local Lyapunov exponents of solar dynamics.
\end{abstract}
\section{Introduction}
\label{sec:Introduction}
The Sun is a mass of incandescent gas \cite{TMBG} whose chief role, as far as life on Earth is concerned, is to fuse hydrogen into helium and produce life--sustaining energy in the process. If that were its only pass--time, however, the Sun would be a luminous, glorified one--trick pony. The Sun also produces neutrinos in its core, exhibits many modes of oscillation throughout its volume, and fires off various ejecta from its surface in the form of electromagnetic energy and plasma. We still do not completely understand all the dynamical processes that occur inside the Sun, but we know that its inner goings--on are intimately connected to our lives in ways biological, spiritual, and technological. We also know that its inner workings are not as simple as \cite{TMBG} implies: it is a fluid sphere whose outer thirty percent rotates differentially while its core rotates more or less uniformly, it exhibits large--scale meridional flows and small--scale turbulence, and it goes through two noteworthy cycles: The 11--year (Schwabe) activity cycle, and the 22--year (Hale) magnetic cycle.
Previous studies (e.g. \cite{Jose:1965} and \cite{Scafetta:2012}) have attempted to explain the variation in solar cycles by appealing to an argument based on planetary tidal forces. The basic line of such an argument is that the gravitational pull on some region of Sun due to all (or some subset) of the planets results in the cyclical nature of solar variability, much as the Moon causes ocean tides on Earth. Apparent similarities between the orbital periods of Jupiter and Saturn, and the 11--year Schwabe cycle make the notion of planetary tides enticing, but certain authors (e.g. \cite{Smythe_Eddy:1977} and \cite{Callebaut_et_al:2012}) have provided strong cases against linear planetary forcing arguments.
In this paper, we propose to apply the joint frameworks of chaos theory and nonlinear time series analysis to solar data in order to further our understanding of the dynamics of the Sun. Naturally, we must begin by answering the question, ``is solar motion chaotic?'' but once we have established that it is, we may move on to the more intriguing challenges of exploring nonlinear connections between chaos in solar motion and other objects in the solar system, and using knowledge of solar dynamics to work toward a method of predicting future solar activity.
\section{History and Recent Work}
\label{sec:History and Recent Work}
Before embarking on a quantitative analysis of chaos in solar activity and dynamics, let us trace the observational history of solar activity through the development of the sunspot record. We will follow this brief history with some examples from the recent literature of how analysis of solar activity is used to benefit the scientific community and society as a whole.
\subsection{Early Work}
\label{sec:Early Work}
Theophrastus of Athens, a student of (and eventual successor to) Aristotle, made the first recorded observation of a sunspot when he identified them as indicators of rain in a meteorological treatise circa 325 BC. In fact, he referred to them casually enough in his writing that we may presume that sunspots were by then already familiar sights \cite{Thomas_Weiss:2008}. At the time of Theophrastus and up until the invention of the telescope in the 17\th century, those interested in the Sun could only observe sunspots during periods of persistent haze or smoke on the horizon. This would have been enough to obscure enough visible light to allow small dark spots on the surface of the Sun to stand out, though it could not have provided any protection from ultraviolet radiation to shield the observer's eyes from long--term harm. In this sense, as would later be true in the case of early studies of radioactive materials, na\"ivit\'e was a boon to science.
The advent of the telescope in Europe meant that scholars and amateur astronomers could more easily track the occurrence rates and motion of sunspots, without having to wait for a sand storm to blow through, or for an entire city to catch fire. It is not surprising that Galileo Galilei had a hand in the sunspot game, but he was not the only interested observer. While he had many contemporary budding heliophiles, Galileo fought most bitterly with Christoph Scheiner over who deserved precedence with regard to the first observation of a sunspot through a telescope, among other matters \cite{Soon_Yaskell:2003}. It is interesting to note that Scheiner first used the transit of sunspots across the solar disk to determine that the Sun's rotational axis is tilted 7.5$^\circ$ from the normal to the ecliptic plane, but when Galileo used the deformation in this tilted transit to argue in favor of a heliocentric solar system, Scheiner (a Jesuit priest) disagreed. Scheiner, for his part, invoked an additional precession of the Sun's axis to maintain the geocentric viewpoint \cite{Thomas_Weiss:2008}. History has borne out Galileo's hypothesis, but unfortunately, the contemporary religious climate did not hold such an opinion in high regard.
William Herschel is credited with developing the first precise past record in 1801 of both the presence and absence of sunspots, and with forwarding the conjecture that an absence of sunspots coincided with higher--than--usual wheat prices in England. For his efforts, the Royal Society of England lavished him with ridicule, though his fate was certainly more pleasant than that of Galileo \cite{Soon_Yaskell:2003}. Given the awe with which the Sun has been regarded for millennia, this certainly could not have been the first time that perceived solar activity was blamed for events on Earth. However, it was the first step toward methodically correlating sunspot activity with climate--related events on Earth (presuming, of course, that the high wheat prices were due to a poor growing season, which was itself due to weather--related phenomena).
Next came Heinrich Schwabe. As is wont to happen during the course of carefully scientific inquiry, Schwabe discovered the sunspot cycle around 1840 when he was searching for something completely different. At the time, he estimated it to be $\sim10$ years. Schwabe had the suspicion that there was an object orbiting the Sun inside the orbit of Mercury, and was therefore making careful observations of any dark spots on the solar disk that would indicate the presence of another body \cite{Thomas_Weiss:2008}. Fortunately for us, sunspots and small planets look very similar, and he was able to make a substantial discovery, despite the fact that it was not the discovery he hoped for.
By the late 1850s, Richard Carrington was stationed at the Red Hill Observatory in Surrey, England, where he recorded occasions in which a dark sunspot would flash white, and a magnetic signature (e.g. the deflection of a compass needle or the interruption of telegraph service) would be registered on Earth \cite{Soon_Yaskell:2003}. In a way, these observations are the logical successors to Herschel's conjecture that sunspot patterns are somehow linked to events on Earth that affect daily life. Carrington and Gustav Sp\"orer independently made two of the most important contributions to heliophysics since Schwabe's discovery of the solar cycle: The first is that sunspots move equatorward throughout the solar cycle; the second is that spots near the equator rotate more quickly than spots at higher latitudes, implying that the Sun rotates differentially, and providing evidence that the Sun is a fluid \cite{Thomas_Weiss:2008}. Sp\"orer is best known for the former (called the ``Sp\"orer Sunspot Law'') and Carrington is best known for the latter (called ``Carrington rotation''). This period also saw Rudolf Wolf, who is best known for establishing a ``relative sunspot number'' (which he used as a way to normalize world--wide sunspot counts), then employing those normalized counts to trace solar cycles back as far as he could. By that time, the heliophysics community had since refined their estimates to place the solar cycle closer to 11 years, so he settled on the cycle from 1755--1766 as Cycle 1. We still employ Wolf's numbering scheme today.
These observations, especially the polar--to--equatorial sunspot drift, set the stage for Edward Maunder at the turn of the twentieth century. Maunder was well aware of Sp\"orer's work and he made a point in his 1904 paper ``Note on the Distribution of Sun--spots in Heliographic Latitude 1874--1902'' to defend Sp\"orer's assertion that the mean sunspot location migrates equatorward against certain members of the scientific community who, Maunder felt, misunderstood Sp\"orer's point \cite{Maunder:1904}. Maunder's conclusions also lent credence to observations made by Wolf around the same time. Figure \ref{fig:MaunderButterflyOrig} reproduces Figure 8 of Maunder's 1904 paper. It is the first occurrence of the ``Butterfly diagram'' method of representing the distribution of sunspots with latitude over the course of one or more solar cycles, an image now very familiar to members of heliophysics community. This representation scheme was one of Maunder's chief contributions to the study of solar activity, but perhaps the most ubiquitous reference to him within the subject of the Sun--Earth connection lies in the name given to the period 1645--1715: ``The Maunder Minimum''. Maunder had identified a dearth of sunspots in telescopic records during this time, but received most of his praise posthumously, when Jack Eddy compiled a thorough review of the data available to him in 1976 and formally named that period of anomalously low solar activity after Maunder \cite{Eddy:1976}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth,height=0.67\textwidth]{/Users/matthewyoung/Desktop/UNH/Courses/thesis/MaunderButterflyOrig.pdf}
\caption{\label{fig:MaunderButterflyOrig} The original image from Maunder's 1904 paper, ``Note on the Distribution of Sun--spots in Heliographic Latitude 1874--1902'' \cite{Maunder:1904}. The winged shape gives this ``Butterfly diagram'' its name. In the diagram, time (in years) moves from left to right while latitude on the solar surface (-40$^\circ$ to +40$^\circ$) moves from bottom to top.}
\end{center}
\end{figure}
\newpage
Maunder was also interested in investigating the relationship between sunspots and geomagnetic activity once studied by Wolf and Edward Sabine in the mid--nineteenth century \cite{Soon_Yaskell:2003}. Whether or not he satisfied his curiosity, it was his contemporary George Hale who took the crucial next step toward connecting sunspots to magnetism. In 1908, Hale discovered Zeeman splitting in spectral lines that he identified with sunspots. In doing so, he determined that the mean magnetic field of a sunspot is $\sim2900$ G (0.29 T) and he proposed that this magnetic field originates in the vortex motion of a sunspot. Since his photographic evidence suggested that all sunspots are vortices, it must follow that all sunspots give rise to a magnetic field \cite{Hale:1908}. The magnetic field intrinsic to a sunspot is not itself the cause of the solar events such as coronal mass ejections (CMEs) and solar flares that disrupt life on Earth, but they are locations on the surface of the Sun out of which the Sun's magnetic field peaks. Thus they were early evidence of the Sun's powerful, persistent magnetic field \cite{Thomas_Weiss:2008}.
With that note, we must make the transition from thinking of sunspots as focal events to thinking of sunspots as proxies for other events more meaningful to life on Earth. The discovery of the sunspot cycle is a testament to good scientific practice, and its analysis over a century and a half has met with wonder and has begotten further interest in heliophysics, but sunspots alone do not drive space weather. Their cycle is, nevertheless, an important indicator of the ebb and flow of solar activity. Most notably, we have come to learn that sunspots are also regions of high--energy activity, and are the seats of solar flares. Solar flares are sources of x--ray radiation that influences Earth's ionosphere and are typically followed by coronal mass ejections (CMEs) that can cause geomagnetic storms if they are directed Earthward \cite{Svesta:2003}
\subsection{Recent Applications}
\label{sec:Recent Work}
Late October into early November 2003 was a period of impressive solar activity referred to within the space science community as the Halloween storms (\cf \cite{Schrijver_Siscoe:2010}, \cite{Hady:2009}). This activity had noticeable electromagnetic effects, including momentarily stripping plasma away from the region of space surrounding the Earth, and disrupting the Earth's magnetic field and producing auroral displays. In addition to those effects on the plasma environment, which are to be expected to some degree, the Halloween storms produced two solar flares that were associated with measurable changes in the density of neutral molecules in the atmosphere \cite{Sutton_et_al:2006}.
Manned space missions pose a threat of high doses of radiation to the astronauts involved, due to the fact that they are no longer shielded from high--energy particles by the Earth's magnetosphere. Mars is a natural destination for manned flights, but the duration of the required trip would cause the flight personnel to be exposed to amounts of hazardous radiation nearly equal to the career limit levels established by NASA \cite{McKennaLawlor_et_al:2012}. Such exposure to background radiation is reasonably well--understood; however NASA still lacks the ability to predict solar flares that could expose astronauts to an additional high dose of radiation (\emph{ibid}).
Beyond the scope of the electromagnetic effects of high--energy bursts of plasma and x--ray radiation, authors have proposed various links between changes in solar activity and changes in human health. Preka--Papadema \emph{et al.} examined procedural records from emergency rooms in two Greek hospitals during scattered months from 2000 through 2006 (Cycle 23). They concluded that cases of cardiological trauma, neurological trauma, and burns, as well as cases related to oncology and pathology increased during periods of heightened solar activity \cite{PrekaPapadema_et_al:2010}. Other authors reported that there is a statistically significant link between certain terrestrial weather cycles, and planetary and solar wind geomagnetic indicators \cite{Halberg_et_al:2010}. Those authors go on to claim that they have found similar cycles ``in over 2500 years of international battles, in 2189 years of tree rings, in around 900 years of the aurora, and in human psychophysiology'' (\emph{ibid}). A third set of authors identify statistically significant periods in cervical epithelial abnormalities matching the approximately 11--year sunspot cycle \cite{Hrushesky_et_al:2011}. They proposal various possible mechanisms, including the effect of changing geomagnetic fields on iron in the human body, flare--related radiation incident upon the surface of the Earth, and subconscious reactions to changes in terrestrial weather induced by space weather events. However, they acknowledge that a correlation between solar cycles and human health cycles does not necessarily mean that a causal effect exists.
As proposed effects of solar activity move further and further from space weather, it is natural to the scientific community (especially the space science community) to greet them with skepticism. However, only a narrow mind will turn away such propositions categorically, and we must not make that mistake. The trouble is that even effects on terrestrial weather are ``indirect'' and thus ``very complex'' \cite{Svesta:2003}, so that a causal link is difficult or impossible to make. That difficulty does not mean we should not continue to study the relationship between solar activity and life on Earth.
\section{Characteristics of Chaotic Systems}
\label{sec:Characteristics of Chaotic Systems}
Thus far, we have considered observations of sunspots and what their relative presence or absence indicates about the current level of solar activity. We have furthermore examined the effect of solar activity on the near--Earth environment, and possible impacts on terrestrial life. However, we have not yet characterized the increase and decrease in solar activity in a precise way. The purpose of this paper is to connect evidence of the variability of solar activity cycles (in the form of sunspot counts) to predictable physical parameters in a quantitative way in order to further our ability to predict the nature of future solar activity.
Simply put, the Sun appears to exhibit characteristics of a chaotic system. Our first task, then, is to precisely define what we mean by the terms ``chaos'' and ``chaotic system''. Deterministic chaos is the motion of a system whose time evolution has a sensitive dependence on initial conditions. We may contrast this with randomness, which describes the motion of a system in which the present state has no causal connection to the previous one \cite{Thornton_Marion:2004}. As a simple example of this contrast, consider the following two dynamic systems: a coin toss, and the waterwheel shown schematically in Figure \ref{fig:ChaoticWaterwheel}. Both systems are subject to the laws of classical physics and both evolve in time according to mathematical formul\ae\ that can be written down. However, the coin toss is random and the waterwheel is chaotic. In the former case, each flip of the coin carries no information about the previous flip, and begets no knowledge of the result of the subsequent flip. There are dynamic quantities at work during each toss (e.g. the angle of the coin relative to the flipper's thumb pre--toss, the weight of coin, etc.), but once the coin lands and the outcome is recorded as ``heads'' or ``tails'', the system resets. The net result of many tosses is subject to the laws of probability alone.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{/Users/matthewyoung/Desktop/UNH/Courses/thesis/ChaoticWaterwheel.pdf}
\caption{\label{fig:ChaoticWaterwheel} Schematic representation of a chaotic waterwheel. The water flows in from the faucet, filling cup the cup beneath, and drains from the bottom of each cup at a specified constant rate. The wheel will begin turning left or right when the weight of water in cup \#1 overcomes the friction forces holding the wheel still. The distribution of water mass will determine the rate and direction of rotation, which will be chaotic for certain rates of flow and drainage.}
\end{center}
\end{figure}
In contrast, the position and velocity of a cup on the waterwheel at time $t_i$ (which, as a pair, are analogous to ``heads'' and ``tails'' after a given coin flip) depend on the position and velocity at time $t_{i-1}$. This should be intuitive, since the matter that makes up the waterwheel can not simply disappear from one location and appear at another, nor can otherwise absent energy suddenly cause a jump in the momentum of that matter. Edward Lorenz found, in the early 1960s, that it is possible to force certain hydrodynamical systems mechanically or thermally (he studied a system that involved both) such that the system exhibits either periodic, quasi--periodic, or non--periodic behavior ``when there is no obviously related periodicity or irregularity in the forcing process'' \cite{Lorenz:1963}. He applied simplifying assumptions to a system in which a cylindrical basin of water was rotated mechanically and subjected to a radial thermal gradient, and derived the following system of equations:
\begin{eqnarray}
\label{eq:LorenzEq1}
\dot{x}&=&-\sigma(x-y) \\
\dot{y}&=&-xz+rx-y \\
\label{eq:LorenzEq2}
\dot{z}&=&xy-bz
\label{eq:LorenzEq3}
\end{eqnarray}
where a dot indicates a time derivative and $\sigma$, $r$, $b$ are parameters. Despite the presence of water and mechanical forcing in the Figure \ref{fig:ChaoticWaterwheel} system, this is not the system that Lorenz studied. What is remarkable, however, is that Equations \ref{eq:LorenzEq1}--\ref{eq:LorenzEq3} exactly describe the chaotic motion of the waterwheel, though they are more commonly expressed as
\begin{eqnarray}
\label{eq:waterwheel1}
\dot{a_1}&=&\omega b_1-Ka_1 \\
\label{eq:waterwheel2}
\dot{b_1}&=&-\omega a_1+q_1-Kb_1\\
\label{eq:waterwheel3}
\dot{\omega}&=&-\left(\frac{\nu}{I}\right)\omega +\left(\frac{\pi g R}{I}\right)a_1
\end{eqnarray}
where $a_1$ and $b_1$ are the cosine and sine Fourier coefficients of the fundamental mode of rotational coordinate of the wheel, respectively, and $\omega$ is the angular velocity of the wheel. We can lend physical insight to the $a_1$ and $b_1$ variables by recalling that $\cos(n\pi x/L)$ and $\sin(n\pi x/L)$ represent the orthogonal basis states for a Fourier decomposition. Thus $a_1$ gives a measure of the ratio of water on the left side of the wheel to that on the right, while $b_1$ gives a measure of the ratio of the water on the top of the wheel to that on the bottom \cite{Benel:2003}.
In Equations \ref{eq:waterwheel1}--\ref{eq:waterwheel3}, $I$ is the moment of inertia of the wheel, $\nu$ is a rotational damping rate, $g$ is the acceleration due to gravity, $R$ is the radius of the wheel, $K$ is the rate of leakage from the cups, $q_1$ is related to the inflow of water, and we have employed the transformation $x\rightarrow \omega/g$, $y\rightarrow (\pi gR/K\nu)a_1$, $z\rightarrow (b_1-q_1/K)/b$ (\cf \cite{Strogatz:1994}). Thus, we have written down a system of differential equations that describe the motion of the waterwheel, and we can use a time--stepping method to solve Equations \ref{eq:waterwheel1}--\ref{eq:waterwheel3} numerically, but we must use techniques from nonlinear stability analysis to make predictions about the asymptotic behavior of the system. What Lorenz found in 1963 is that there exist values for the parameters in his original system of equations that precluded stable fixed points and limit cycles, but for which the system's trajectory remained in a bounded set \cite{Lorenz:1963}. Such behavior is a hallmark of chaos.
A more illustrative example of chaos is the strange attractor. In short, the strange attractor is that bounded set in phase space on which a system's trajectory remains. More precisely, a strange attractor is an attractor which exhibits sensitive dependence on initial conditions, where an \emph{attractor} is a minimal invariant set that attracts an open set of initial conditions. We can decode this definition by separating it into three terms: To attract an open set of initial conditions means that when a trajectory starts out in some open set that contains the attractor, that trajectory gets arbitrarily close to the attractor as $t\rightarrow\infty$. The word invariant means that once a trajectory is on the attractor, it does not leave, and the word minimal means that there is no proper subset of the attractor's set that is both invariant and attracts an open set of initial conditions \cite{Strogatz:1994}.
In addition to the definition of a strange attractor given above, strange attractors can be distinguished from the attractors that describe linear processes by their fractal, or non--integer, dimension. In short, fractal dimension is an extension of the more familiar integer dimension to geometric figures for which the density of points around a given point changes over the figure. What we define as the fractal dimension of a strange attractor (which, incidentally, is approximately 2.06 for the Lorenz system) is a sort of average of the ``local dimension'' over the entire attractor \cite{Abarbanel:1996}.
The strange attractor for the Lorenz Equations is shown four ways in Figure \ref{fig:LorenzAtt_views}: In the upper left panel, we show the attractor in three dimensions, and in each of the other three panels, we show the attractor projected onto one of the three possible planes. The motivation for showing the Lorenz attractor in this way is to illustrate its complexity as a geometrical object, as well as to demonstrate the risk in projecting an attractor onto too--low a dimension: Any of the two--dimensional images could appear to be a legitimate description of the dynamics, but the three--dimensional images shows that we need more than two dimensions to truly capture the dynamical trajectories. We are fortunate to have the differential equations for the Lorenz system, since knowledge of the underlying equations allows us to compute quantities such as the fractal dimension analytically. In most experimental applications, the only data are from a noisy time series sampled with finite precision for a finite length of time. In those cases, we must resort to numerics if we wish to characterize the attractor and move forward with analysis.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\textwidth,height=0.5\textwidth]{/Users/matthewyoung/Desktop/UNH/Courses/thesis/LorenzAtt_views.pdf}
\caption{\label{fig:LorenzAtt_views} The strange attractor formed by the Lorenz Equations (equations \ref{eq:LorenzEq1}--\ref{eq:LorenzEq3}), with $\sigma=10$, $b=8/3$, and $r=28$. The upper left--hand panel shows a 3--D box plot of the set, while the other three panels show each of the three possible planes.}
\end{center}
\end{figure}
Fractal dimension is considered one of two invariants of the motion; the second is a class of numbers called global Lyapunov exponents. In much the same way as we can characterize a linear system by its Fourier spectrum, we can characterize a chaotic system by its fractal dimension and its global Lyapunov exponents \cite{Abarbanel:1996}. In chaotic systems, points on an attractor diverge or converge exponentially quickly \cite{Strogatz:1994}. We know that a strange attractor is an attractor that exhibits sensitive dependence upon initial conditions. Thus, given an $N$--dimensional sphere of initial conditions, we can demand that the radius of the sphere be infinitesimally small and we will still have initial conditions that differ from each other enough that the sphere spreads out into an $N$--dimensional ellipsoid as time evolves. We can describe the time--dependent $k^{th}$--principle axis of the ellipsoid by $\delta_k(t)\sim\delta_k(0)e^{\lambda_kt}$ (we may consider $\delta_k(0)$ the distance of the $k^{th}$ initial condition from the center of the initial sphere) so that if at least one $\lambda_k$ is positive, that principle axis will stretch the sphere in the $k^{th}$ direction. Note that this exponential time dependence appears here as an ansatz, but is in fact predicted by Oseledec's multiplicative ergodic theorem \cite{Kelliher:2011}. If time evolves for long enough, the ellipsoid of initial conditions will be so stretched out over the attractor that we will have reached a point at which prediction breaks down \cite{Strogatz:1994}. We shall revisit the concept of Lyapunov exponents in $\S$\ref{sec:Nonlinear Analysis of Time Series Data}
Now we can be exact when we speak of sunspot counts and solar activity as chaotic. When we hypothesize that solar activity follows a chaotic cycle, we imply that the magnitude of the current cycle depends in some way on the previous cycle or previous cycles, that the evolution through time of the solar cycle is sensitive to conditions at an earlier time, and that we can find an attractor that contains the information we need to understand the relevant dynamics. We are interested in describing the chaotic evolution of sunspot counts and other dynamical variables related to the Sun in order to determine which (possibly constant or periodic) physical drivers may be responsible for chaotic variations in solar activity, and then to use that knowledge to predict solar activity as far in the future as possible.
\section{Nonlinear Analysis of Time Series Data}
\label{sec:Nonlinear Analysis of Time Series Data}
When presented with a time series that does not exhibit periodic behavior, we can still attempt to analyze the underlying dynamics by assuming that the data posses a nonlinear dependence. There exists a systematic method for uncovering dynamical dependence on multiple time scales known as attractor reconstruction, though determination of the correct value of certain parameters is not always simple. In order to suss out nonlinear time dependence from a vector $x(t)$ of scalar values sampled at discrete time points (i.e. a time series), we construct the $N$--dimensional vector
\[
\mathbf{y}(t)=(x(t),x(t+\tau),...,x(t+(N-1)\tau))
\]
Therefore, we must know \emph{a priori} the appropriate values of $N$ and $\tau$ with which to construct $\mathbf{y}(t)$ (\cf \cite{Strogatz:1994} and \cite{Abarbanel:1996}, but note that notations among this work and those two works differ). If the ``embedding dimension'' $N$ is chosen to be too small, we can not construct $\mathbf{y}(t)$ such that it completely unfolds the dynamics of $x(t)$; if we choose $N$ to be too large, noise will appear to be linked to the dynamics \cite{ChunHua_XinBao:2008}. On the other hand, if we choose the ``time delay'' $\tau$ to be too small, the resultant components of $\mathbf{y}(t)$ will not be distinct enough to give us information about how the system evolves; if we choose $\tau$ to be too large, the intrinsic instability of the dynamics will cause $x(t)$ and $x(t+\tau)$ to act like random data points with respect to each other \cite{Abarbanel:1996}. This intrinsic instability is, of course, a result of a positive maximum global Lyapunov exponent.
In our discussion of Lyapunov exponents above (see $\S$\ref{sec:Characteristics of Chaotic Systems}), we claimed that if at least one Lyapunov exponent is positive, and if time evolves ``long enough'', arbitrarily--close initial conditions will diverge so much that predictions based on the original information will break down. This amounts to two limiting processes: one as time goes to infinity, and the other as distance between nearby initial conditions goes to zero. Mathematically, we have
\begin{equation}
\lambda_j=\displaystyle{ \lim_{t \to \infty} \lim_{\delta_0 \to 0}}\left(\frac{1}{t}\left| \frac{\delta(x_j,t)}{\delta_0} \right|\right)
\label{eq:GLE}
\end{equation}
Equation \ref{eq:GLE} is a \emph{definition}, but it is not a very useful prescription for calculating the spectrum of global Lyapunov exponents from time series data. In effect, what we are doing when we calculate Lyapunov exponents is examining the stability of the dynamics to perturbations. Given a map $\mathbf{f}(\mathbf{y})$ that takes a point on the attractor $\mathbf{y}_i$ to the point $\mathbf{y}_{i+1}$, we consider a perturbation to first order:
\begin{eqnarray}
\nonumber
\mathbf{y}_{i+1}+\mathbf{\delta}_{i+1}&=&\mathbf{f}(\mathbf{y}_i+\mathbf{\delta}_i)\approx\mathbf{f}(\mathbf{y}_i)+\sum\limits_{a,b}\frac{\partial f_a(\mathbf{y}_i)}{\partial x_b}\cdot\mathbf{\delta}_i \\
\mathbf{\delta}_{i+1}&\approx&\sum\limits_{a,b}\frac{\partial f_a(\mathbf{y}_i)}{\partial x_b}\cdot\mathbf{\delta}_i
\label{eq:perturb}
\end{eqnarray}
where the last simplification comes from the definition of the map $\mathbf{f}(\mathbf{y}_i)=\mathbf{y}_{i+1}$ \cite{Abarbanel:1996}. Thus, to first order, we are actually interested in the Jacobian matrix of $\mathbf{f}(\mathbf{y}_i)$. After applying this map a large number (call it $n$) of times, we can expect to learn about the stability of nearby initial conditions at long times. This last process relies on composing the Jacobian matrix with itself $n$ times and forming the following matrix prescribed by the Oseledec multiplicative ergodic theorem mentioned in $\S$\ref{sec:Characteristics of Chaotic Systems}
\begin{equation}
\mathbf{O}=\left\{\left[\sum\limits_{a,b}\frac{\partial f_a(\mathbf{y}_i)}{\partial x_b}\right]^T\cdot\sum\limits_{a,b}\frac{\partial f_a(\mathbf{y}_i)}{\partial x_b}\right\}^{1/2n}
\label{eq:Oseledec}
\end{equation}
Finally, the logarithms of the eigenvalues of $\mathbf{O}$ as $n\rightarrow\infty$ are the global Lyapunov exponents \cite{Abarbanel_et_al:1991}, \cite{Abarbanel:1996}.
What if $n$ is not large? can we will learn information about the dynamics from the ratio inside the parentheses in Equations \ref{eq:GLE} or from the logarithms of the eigenvalues of $\mathbf{O}$? The answer is yes. These quantities, for finite time or $n$ not large, are called the local Lyapunov exponents. Abarbanel \emph{et al.} found that local Lyapunov exponents are relevant for predicting short--term dynamics or for characterizing the average variation of Lyapunov exponents over the attractor, as opposed to describing the long--term evolution of trajectories on the attractor \cite{Abarbanel_et_al:1991}. The local Lyapunov exponents also provide a measure of the ``heterogeneity'' of the system in that they can characterize short--term chaotic behavior in an otherwise non--chaotic system \cite{Bailey:1997}.
\section{The Solar Dynamo}
\label{sec:The Solar Dynamo}
Early analysis of the inner dynamics of the Sun focussed on hydrodynamic effects. Csada considered the fluid motion of the Sun, including differential rotation and meridional circulation, but neglected electromagnetic effects \cite{Csada:1949}. However, research has since shown that the electromagnetic induction that arises via differential rotation is the most effective mechanism for generating the toroidal fields seen in the solar dynamo \cite{Rudiger:1989}. In what follows, we will refer loosely to four main spherically--symmetric regions ($R_S$ denotes the radius of the Sun):
\begin{inparaenum}
\item the radiative zone, extending from the core out to $\sim0.7R_S$, in which rotation is thought to be latitudinally uniform,
\item the convective zone, which lies between the radiative zone and the photosphere at $\sim1R_S$ and exhibits latitudinally--varying rotational rates (differential rotation),
\item the tachocline, which lies at the interface between the radiative and convective zones, and
\item the overshoot layer, in which the convective zone spills briefly over into the radiative zone.
\end{inparaenum}
Figure \ref{fig:sunSchematic} shows a cartoon schematic of the inside of the Sun.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.6\textwidth,height=0.6\textwidth]{/Users/matthewyoung/Desktop/UNH/Courses/thesis/sunSchematic.pdf}
\caption{\label{fig:sunSchematic} Cartoon representation of some major features of the Sun. The radiative zone extends out to $\sim0.7R_S$, with the convective zone covering the balance out to the photosphere. The tachocline (not labeled) sits at the interface between the radiative and convective zones and the overshoot layer lies just below the tachocline. Image credit: Encyclopedia of the Earth (\url{http://www.eoearth.org/article/Solar_radiation})}
\end{center}
\end{figure}
The early theory of Parker is regarded as seminal work in the development of solar dynamo theory \cite{Parker:1955a}, \cite{Parker:1955b}. He acknowledged that the single relationship between a magnetic field and a fluid flow given by
\begin{equation}
\frac{\partial\mathbf{B}}{\partial t}=\nabla\times(\mathbf{v}\times \mathbf{B})+\frac{1}{\mu\sigma}\nabla^2\mathbf{B}
\label{eq:ParkerEq1}
\end{equation}
can not maintain both a poloidal magnetic field and a toroidal magnetic field, but found that an existing poloidal field \emph{can} generate a toroidal field that would then regenerate the poloidal field through Equation \ref{eq:ParkerEq1} \cite{Parker:1955b}. Given a poloidal magnetic field generated by a rotating conducting core, differential shearing can deform field lines by stretching them in the azimuthal direction, thereby generating a toroidal magnetic field (\cf \cite{Rudiger:1989} and \cite{Nandy:2010}). Once this toroidal field has been established, radially--rising cyclones in the radiative zone push radial field lines outward until they exhibit a non--axisymmetric bulge (imagine the capital letter omega: $\Omega$). The cyclonic motion of these convective structures twists the bulged field line to give it a component parallel to the meridional plane, thereby regenerating the poloidal magnetic field \cite{Parker:1955b}. We have come to refer to the generation of toroidal magnetic fields via differential rotation as ``the $\Omega$ effect'' (this is in reference to the differential rotation, and is not to be confused with the $\Omega$ used to visualize radial stretching of toroidal field lines), and to the generation of poloidal magnetic fields via twisting of toroidal field lines as ``the $\alpha$ effect'' \cite{Hotta_Yokoyama:2010}.
We have seen that the cyclonic flows proposed by Parker are crucial for generating the toroidal magnetic field at the interface between the radiative and convective zones, but the convective zone proper has its own characteristic flow that supports the solar dynamo. Choudhuri \emph{et al.} developed a model that coupled the solar surface to the low--latitude base of the solar convective zone via large--scale meridional circulation \cite{Choudhuri_et_al:1995}. Observational evidence suggests poleward meridional flows at the surface, while mass conservation suggests closure through equatorward flows at the base of the convective zone \cite{Choudhuri:2003}. They noted that when the meridional circulation occurred on a timescale shorter than that of magnetic diffusion through the convective zone, rising loops of buoyant toroidal flux (a topic to be addressed in greater detail below) would produce butterfly diagrams in qualitative agreement with observation, and postulated that this effect is due to overcoming constraints imposed by coupling the two layers simply through diffusion. A decade later, \cite{Jiang_et_al:2007} provided evidence that the solar dynamo is a diffusive dynamo. The meridional flow also acts upon decaying bipolar sunspot pairs to cause a net magnetic flux to move poleward, which, in turn, reverses the older polar field and builds up the poloidal field for the next cycle \cite{Nandy:2010}.
One final piece of the solar magnetic puzzle is the diffuse field that exists in the regions \emph{not} occupied by sunspots. We will not consider this diffuse field here. However, the last few years of model and observational evidence have shown that this diffuse magnetic field gives important information about the dynamo process \cite{Choudhuri:2003}. Therefore, we would be remiss to not mention it at all.
Now that we have established a foundation for understanding the solar dynamo process, we move onto sunspot formation proper. To begin our consideration of sunspot formation, we return to Parker, who remarked even in 1955 that the fact that overall visible changes in the corona are in step with magnetic activity suggests that ``sunspots and prominences are not just individual isolated magnetic phenomena but are secondary effects of a general solar magnetic cycle'' \cite{Parker:1955b}. In fact, he had already acknowledged, earlier the same year, that if a closed loop of magnetic flux (hereafter referred to as a ``flux loop'' or ``flux tube'') contained in the toroidal magnetic field is in pressure equilibrium, then the external fluid pressure should be in balance with both the internal fluid pressure and the internal magnetic pressure \cite{Parker:1955a} (\cf \cite{Nandy:2010}):
\begin{equation}
p_{ext}=p_{int}+\frac{B^2}{8\pi}
\label{eq:magPressure}
\end{equation}
Thus, there will be a net (hydrostatic) buoyant force on the flux loop since $p_{int}