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%\chapter{Chaos and predictability}

In this chapter we study the structure of the strange attractor by use of the well-known chaos model of Lorenz(1963), and perform some basic experiments concerning the predictability.

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\section{Theoretical Background}

We assume that $X=X(t)$ represents the intensity of the convective motion, $Y=Y(t)$ represents the temperature difference between the ascending and the descending current, and $Z=Z(t)$ represents the deviation of the vertical temperature profile from linearity.
By approximating extremely the partial differential equations of the heat convection, Edward Lorenz derived the following set of the simple three ordinary differential equations,
\begin{align}
  \frac{d X}{d t} =& - \sigma X + \sigma Y \\
  \frac{d Y}{d t} =& - X Z + \gamma X - Y \\
  \frac{d Y}{d t} =& X Y - b Z
\end{align}
where $\sigma$, $\gamma$, and $b$ denote the Prandtl number, the Rayleigh number, and the spatical periodicity of the convection, respectively.

Lorenz discovered {\it chaos} by performing time-integration of the above equations numerically in the old days when the computer had not developed enough.
He expressed chaos as 'an irregular motion under determinism' in his monumental paper (Lorenz, 1963).

Lorentz (1963) described the numerical solution at $\sigma =10, b=8/3, \gamma =28$.
Assuming $\sigma$ as 10 and $b$ as 8/3, we study the dependence of the stability of the solution on $\gamma$.
For $0 \leq \gamma \leq 1$, the steady state is only the heat conduction (Berge et al., 1984),
\begin{equation}
X = Y = Z =0.
\end{equation}

For $\gamma$ slightly larger than 1, the solution of the heat conduction becomes unstable, while other two solutions arise
\begin{equation}
X = Y = \pm [ b ( \gamma - 1) ]^{1/2}, \quad Z = \gamma -1.
\end{equation}
They represent the steady convection.
Although these two solutions are linearly stable for $1 \leq \gamma \leq 24.74$, they become unstable for $\gamma = 24.74$, and for $\gamma > 24.74$ only nonperiodic solutions exist.

In more detail, for $ 24.06 < \gamma < 24.74$ the three {\it atrractors} (a region attracting a trajectory in {\it phase space}) coexist so that {\it hysteresis} is observed.
One of them is a {\it strange atrractor}, and the others are the steaty heat convection already mentioned.
That strange attractor which appears for $ \gamma < 24.74$ is called as a {\it non-standard} strange attractor, and chaos exists quasi-stable ({\it meta stable chaos}) for $\gamma > 13.926$.
For $ 24.74 < \gamma < 30.4$, only the strange attractor exists stable.
It may be the reason why many numerical studies have done at $\gamma = 28$.

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\section{Exercises}

Perform exercises about Lorenz Chaos by using the programs "C12-1", "C12-2", and "C12-3".
By dragging the graphic window as shown in Figure~\ref{fig:C13_01BC}, we can rotate the result of both "C12-1" and "C12-2" and observe from various view angles.
These programs use the Runge-Kutta-Gill method for numerical integration.

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C13_01.epsf}
    \caption{Result of "C12-1" using parameters of Example 1.}
    \label{fig:C13_01A}
  \end{center}
%\end{figure}
%\begin{figure}
  \begin{center}
\[
  \begin{array}{cc}
    \includegraphics[width=5cm,keepaspectratio,clip]{fig/C13_02.epsf} &
    \includegraphics[width=5cm,keepaspectratio,clip]{fig/C13_03.epsf}
  \end{array}
\]
%    \includegraphics[width=5cm,keepaspectratio,clip]{fig/C13_01B.epsf} \\
%    \includegraphics[width=5cm,keepaspectratio,clip]{fig/C13_01C.epsf}
    \caption{Samples rotating the result of "C12-1" shown in Figure~\ref{fig:C13_01A}.}
    \label{fig:C13_01BC}
  \end{center}
\end{figure}

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\subsection{Exercise 1 - the Lorenz attractor}

Set the parameters as the same as the Lorenz's experiment ($\sigma =10, b=8/3, \gamma = 28$, the initial condition $(X,Y,Z) = (0,1,0)$), and view a three-dimensional image of a strange attractor from various angles to comprehend its characteristic structure.
Note that the parameter $b$ is expressed as \texttt{f/g} in the program.

\begin{description}
\item[Example 1] The standard experiment.
\end{description}
\begin{center}
\begin{tabular}{ccccccc}
  10.0 & 28.0 & 8.0 & 3.0 & 0.0 & 1.0 & 0.0 \\
  $\sigma$ & $\gamma$ & f & g & X & Y & Z
\end{tabular}
\end{center}
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\subsection{Exercise 2 - various strange attractors}

In this section, display various strange attractors by changing the parameters ( $\sigma , b, \gamma , X, Y, Z$ ), and  perform some applied experiments to have a profound understanding of chaos itself.
The programs used here are \texttt{lore3} and \texttt{lore4}.

\begin{itemize}
\item "C12-2" shows sharp dependence of chaos on the initial state. Lorenz(1997) explained {\it sharp dependence on the initial state} as follows.
Two states subjected to determinism will keep identical if the current states are perfectly identical.
On the other hand, if there is a slight difference between the current states, the difference will grow with the course of time, so that the two states will be quite different.
We may say that phenomena with this nature, {\it sharp dependence on the initial state}, are {\it chaotic}.
\item "C12-3" displays a {\it Lorenz plot}. The abscissa is $Z_n$, the value of the $n$th maximum of a variable Z, while the ordinate is $Z_{n+1}$, the value of the following maximum. Each point represents a mapping from $Z_n$ to $Z_{n+1}$. Lorenz idealized this Lorenz plot as a two-to-one correspondence, and examine its property to argue the nature of chaos.
\end{itemize}

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C13_04.epsf}
    \caption{Result of "C12-2" using parameters of Example 1.}
    \label{fig:C13_04}
  \end{center}
\end{figure}

\begin{figure}[t]
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C13_05.epsf}
    \caption{Result of "C12-3" using parameters of Example 1.}
    \label{fig:C13_05}
  \end{center}
\end{figure}

After running both "C12-2" and "C12-3" using the previous parameters, change the constants $\sigma$, $\gamma$, and $b$ to examine the parameter dependence of solutions of the system.
%Note that the constant $\sigma$ is the Prandtl number ( the ratio of the kinematic viscosity to the thermal diffusivity ) and that the value of $\sigma =10.0$ is comparable to a value of water.
Note that $\sigma =10.0$, where $\sigma$ is the Prandtl number, corresponds to a value of water.
If you want to perform the experiments concerning a convection in the air, change the value of $\sigma$ as $1.0$. 

\begin{description}
\item[Example 2] Set the parameters as follows,
\end{description}
\begin{center}
\begin{tabular}{ccccccc}
  10.0 & 28.0 & 8.0 & 3.0 & 30.0 & 30.0 & 30.0 \\
  $\sigma$ & $\gamma$ & f & g & X & Y & Z
\end{tabular}
\end{center}
%Change the initial state $(X, Y, Z)$ and run the programs.
%At some value of $\sigma$, either a steady state or a strange attractor appears dependent on the initial state.
%Consider the reason why .
Perform the programs with another set of parameters in $(X, Y, Z)$ which corresponds to the initial state.
Consider the reason why either a steady state or a strange attractor appears dependent on the initial state even we assume the same value of $\sigma$.

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\section{Appendix}

Lorenz (1997,p10) said as follows.

\begin{quotation}
'Under any system sharply dependent on the initial state, it is impossible to predict the sufficiently distant future even at a mediocre level, unless perfectly accurate observations are available.'
\end{quotation}

In weather forecasting, {\it daily forecast} is one of the problems due to the sensitivity on the initial state; this sensitivity is arising from a chaotic nature of the atmosphere.
In medium-range (1 to 2 weeks) forecasting at mid-latitudes, we often find that a predicted state changes significantly at every update of the initial state, and a next prediction is not always better than a previous one (Tsuyuki 1999).
In recent years, this is one of the serious problems for improving the accuracy of medium-range forecasting.
It will be interesting to cite some methods adopted today to handle this problem and consider their merits.

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\section*{Reference}

\begin{description}
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\item E. N. Lorenz, 1963 : Deterministic Nonperiodic flow, {\it J. Asmos. Sci.}, {\bf 20}, 130-141
\item P. Berge, Y.Pomeau, C.Vidal, 1984 : L'ordre dans le chaos - vers une approche deterministe de la turbulence
\item E. N. Lorenz, 1996 : The Essence of Chaos (The Jessie and John Danz Lecture Series), {\it Univ. of Washington Press}
\item Tsuyuki, 1999 : Data assimilation and sensitivity on the initial state, {\it Tenki}, {\bf 46-3}, 179-184 (in Japanese)
\end{description}

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