\sectionhead{A Discrete Nonlinear System}
     \begin{itemize}
          \item  Having spent some time looking at {\em linear} systems, now it's time to move on to {\em nonlinear}
               systems.
               
               We will start with a classic example, called the discrete {\em Logistic Equation}.  This example is very simple
               to develop and specify, but in its details it reveals many of the characteristic properties of nonlinear
               dynamical systems.
     \end{itemize}
\pagedone
\subsectionhead{The Logistics Equation (derivation)}
     \begin{itemize}
          \item Now it's time to get down to business, and start exploring a specific nonlinear example.
               We're going to look at a classic example from biology, concerning population growth.
          
               We'll start by considering a single
               species, living in an environment where it depends on a consumable resource.  So, imagine a beaker
               of sugar water, into which we put a single bacterium:
           \begin{center}
                \begin{tikzpicture}[line width=1.75pt, color=gray]
                     \draw[rounded corners]
                            (-1.5,3) -- (-1.3,3) -- (-1.3,0) -- (1.3,0) -- (1.3,3) -- (1.5,3);
                     \draw (0,2.4) ellipse (1.3cm and 0.2cm);
                     \draw (0,1) circle(.05cm);
%                     \filldraw
%                           [yscale=2.8] (-1.3,1)
%                           [rounded corners] -- (-1.3,0) -- (1.3,0)
%                           [sharp corners] -- (1.3,1) -- cycle;
                \end{tikzpicture}
           \end{center}
 \pagedone
          As time passes, the bacteria reproduce (by dividing, time lapse pictures, bacteria greatly magnified \ldots):
          \begin{center}
                \begin{tikzpicture}[line width=1.75pt, color=gray]
                     \draw[rounded corners]
                            (-9.5,3) -- (-9.3,3) -- (-9.3,0) -- (-6.7,0) -- (-6.7,3) -- (-6.5,3);
                     \draw (-8,2.4) ellipse (1.3cm and 0.2cm);
                     \foreach \x in {0.0}
                          \foreach \y in {0.0}
                               \draw (-8+\x,1+\y) circle(.05cm);
                     \draw(-8,-1)node{$P_0$};
                     \draw[rounded corners]
                            (-5.5,3) -- (-5.3,3) -- (-5.3,0) -- (-2.7,0) -- (-2.7,3) -- (-2.5,3);
                     \draw (-4,2.4) ellipse (1.3cm and 0.2cm);
                     \foreach \x in {0.1, -0.1}
                          \foreach \y in {0.0}
                               \draw (-4+\x,1+\y) circle(.05cm);
                     \draw(-4,-1)node{$P_1$};
                     \draw[rounded corners]
                            (-1.5,3) -- (-1.3,3) -- (-1.3,0) -- (1.3,0) -- (1.3,3) -- (1.5,3);
                     \draw (0,2.4) ellipse (1.3cm and 0.2cm);
                     \foreach \x in {0.1, -0.1}
                          \foreach \y in {0.1,-0.1}
                               \draw (0+\x,1+\y) circle(.05cm);
                     \draw(0,-1)node{$P_2$};
                     \draw[rounded corners]
                            (2.5,3) -- (2.7,3) -- (2.7,0) -- (5.3,0) -- (5.3,3) -- (5.5,3);
                     \draw (4,2.4) ellipse (1.3cm and 0.2cm);
                     \foreach \x in {0.3, 0.1,-0.1,-0.3}
                          \foreach \y in {0.1,-0.1}
                               \draw (4+\x,1+\y) circle(.05cm);
                     \draw(4,-1)node{$P_3$};
                     \draw[rounded corners]
                            (6.5,3) -- (6.7,3) -- (6.7,0) -- (9.3,0) -- (9.3,3) -- (9.5,3);
                     \draw (8,2.4) ellipse (1.3cm and 0.2cm);
                     \foreach \x in {0.3, 0.1,-0.1,-0.3}
                          \foreach \y in {0.3,0.1,-0.1,-0.3}
                               \draw (8+\x,1+\y) circle(.05cm);
                     \draw(8,-1)node{$P_4$};
%                     \filldraw
%                           [yscale=2.8] (-1.3,1)
%                           [rounded corners] -- (-1.3,0) -- (1.3,0)
%                           [sharp corners] -- (1.3,1) -- cycle;
                \end{tikzpicture}
           \end{center}
           The population of bacteria at time $n+1$ will be given by
                $$P_{n+1} = 2 P_n \ \ (\text{with\ }P_0 = 1),$$
           and so in general
                $$P_n = 2^n.$$
\pagedone
          \item Now we'll ask the traditional question at this point:  If we put the first bacterium in the beaker at 9:00 am,
          the bacteria divide every 10 minutes, and the beaker is completely full of bacteria at 12:00 noon, at what time
          was the beaker exactly half full of bacteria?
          
          The answer is clearly ``11:50 am.''  (Now we're supposed to think about natural resources like oil, and oil consumption
          doubling every 20 years -- if in all the time of oil consumption up until now we have consumed half the total reserve
          of oil, in how many years will we have consumed all the oil? etc. \ldots)
          
          But the question I really want to ask is, ``What will the beaker look like at 10 minutes after 12:00 noon?''
          
          The first thought is that there will be sugar water and bacteria all over the table because the beaker will have
          overflowed from the next doubling (it was completely full at 12:00 noon) -- but in the real world, that can't be
          right.  In fact, there won't be any live bacteria in the beaker at 10 minutes after 12:00 noon, because they
          all will have died of starvation!  The beaker was completely full of bacteria, so there was no sugar water
          left \ldots
          
          In fact, as the bacteria population gets closer to filling the beaker, there will be a downward pressure on
          population -- there will be increasing competition for scarce resources.  Therefore, a better model of
          the bacteria population at a given time $n+1$ will be
               $$ P_{n+1} = 2 P_n (M - P_n)$$
          where the 2 is from population growth by doubling, and $M$ is the maximum number of bacteria that can
          fit in the beaker.  Note that this corresponds with the idea that if the beaker is ever completely full of
          bacteria, in the next time step the population will go to $0$ because there will be mass starvation \ldots
          
     \item     Now let's simplify the units -- instead of keeping track of the total population $P_n$, we'll let $x_n = \frac{P_n}{M}$,
          that is, the proportion of the maximum population (also sometimes called the {\em carrying capacity}) that we have
          at time $n$.  We will thus have $0 \le x_n \le 1$.  Let's also generalize to other species that might have
          a net birth rate $R$ other than 2, so that
          we will have the classic {\em logistics equation} for population of a single species in a resource limited
          environment:
               $$x_{n+1} = R * x_n * (1 - x_n).$$
               
          Note that this has an $x_n^2$ term, is thus nonlinear, and can't be ``solved'' in any straightforward way \ldots
          
          
\end{itemize}
\pagedone
\subsectionhead{The Logistics Equation (analysis)}
     \begin{itemize}
          \item  The fact that we can't ``solve'' the logistics equation doesn't mean that we can't study its behavior,
               or analyze its characteristics.  Let's do some work on that.  We can start by just observing the system
               in action.
               
               For our first example, let's look at what happens when $R = 1$.  We'll choose a starting value of $x_0 = \frac{1}{2}$.
               We will have
               \begin{align*}
                    x_0 &= \frac{1}{2} \\
                    x_1 &= 1 * \frac{1}{2} * (1 - \frac{1}{2}) = \frac{1}{4} \\
                    x_2 &= 1 * \frac{1}{4}(1 - \frac{1}{4}) = \frac{1}{4} * \frac{3}{4} = \frac{3}{16}\\
                    \ldots
               \end{align*}
\pagedone
               Plotting $x_n$ vs. $n$, we see (first $x_0=\frac{1}{2}$, then $x_0=0.75$):
               \begin{center}
                    \vspace{-0.5cm}
                    \begin{tikzpicture}[line width=1.75pt]
                          \draw[color=black, ->] (-8.2,0) -- (8,0);
                          \draw[color=black](-8.6,2) node{$x_n$};
                          \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                          \draw[color=black](-8.6,4) node{$1$};
                          \draw[color=black](-1,-1) node{$x_{n+1} = 1.0 * x_n *(1 - x_n),\ \ x_0=\frac{1}{2}$};
                          \draw[color=black](-8.2,4) -- (-7.8,4);
                          \draw[color=black](8.4,0) node{$n$};
                          \pgfmathsetmacro{\R}{1.0}
                          \pgfmathsetmacro{\x}{0.5}
                          \draw[fill=black](-8,4*\x) circle(0.05cm);
                          \foreach \i in {-7.5,-7,...,7} {
                               \pgfmathparse{\R*\x*(1-\x)}
                               \xdef\x{\pgfmathresult}
                               \draw[fill=black](\i,4*\x) circle(0.05cm);
                          }
                    \end{tikzpicture}
               \end{center}
%                \begin{center}
%                    \vspace{-0.5cm}
%                    \begin{tikzpicture}[scale=4,line width=1.75pt]
%                          \draw[color=black, ->] (0,-0.05) -- (0,1.1);
%                          \draw[color=black](-0.15,0.5) node{$x_n$};
%                          \draw[color=black, ->](-0.05,0.0) -- (4,0.0);
%                          \draw[color=black](-0.15,1) node{$1$};
%                          \draw[color=black](1.75,-0.25) node{$x_{n+1} = 1.0 * x_n *(1 - x_n),\ \ x_0=\frac{1}{2}$};
%                          \draw[color=black](-0.05,1) -- (0.05,1);
%                          \draw[color=black](4.1,0) node{$n$};
%                          \pgfmathsetmacro{\R}{1.0}
%                          \pgfmathsetmacro{\x}{0.5}
%                          \draw[fill=black](0,\x) circle(0.0125cm);
%                          \foreach \i in {0.125,0.25,...,3.75} {
%                               \pgfmathparse{\R*\x*(1-\x)}
%                               \xdef\x{\pgfmathresult}
%                               \draw[fill=black](\i,\x) circle(0.0125cm);
%                          }
%                    \end{tikzpicture}
%               \end{center}
               \begin{center}
                    \vspace{-0.5cm}
                    \begin{tikzpicture}[line width=1.75pt]
                          \draw[color=black, ->] (-8.2,0) -- (8,0);
                          \draw[color=black](-8.6,3) node{$x_n$};
                          \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                          \draw[color=black](-8.6,4) node{$1$};
                          \draw[color=black](-1,-1) node{$x_{n+1} = 1.0 * x_n *(1 - x_n),\ \ x_0=0.75$};
                          \draw[color=black](-8.2,4) -- (-7.8,4);
                          \draw[color=black](8.4,0) node{$n$};
                          \pgfmathsetmacro{\R}{1.0}
                          \pgfmathsetmacro{\x}{0.75}
                          \draw[color=black,fill=black](-8,4*\x) circle(0.05cm);
                          \foreach \i in {-7.5,-7,...,7} {
                               \pgfmathparse{\R*\x*(1-\x)}
                               \xdef\x{\pgfmathresult}
                               \draw[color=black,fill=black](\i,4*\x) circle(0.05cm);
                          }
                    \end{tikzpicture}
               \end{center}
\pagedone
          Let's look at some other values of $R$ (and values of $x_0$):
          \begin{center}
               \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 2.0 * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{2.0}
                     \pgfmathsetmacro{\x}{0.5}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \begin{center}
         %      \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,3) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 2.0 * x_n *(1 - x_n),\ \ x_0=0.75$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{2.0}
                     \pgfmathsetmacro{\x}{0.75}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
\pagedone
          \begin{center}
          %     \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 3.0 * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{3.0}
                     \pgfmathsetmacro{\x}{0.5}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \begin{center}
             %  \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 3.5 * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{3.5}
                     \pgfmathsetmacro{\x}{0.5}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
\pagedone
          \begin{center}
           %    \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 3.6 * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{3.6}
                     \pgfmathsetmacro{\x}{0.5}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \begin{center}
            %   \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 3.7 * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{3.7}
                     \pgfmathsetmacro{\x}{0.5}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
\pagedone
          \begin{center}
           %    \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 3.8 * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{3.8}
                     \pgfmathsetmacro{\x}{0.5}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \begin{center}
          %     \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 3.9 * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{3.9}
                     \pgfmathsetmacro{\x}{0.5}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
\pagedone
          \begin{center}
           %    \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 4.0 * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{4.0}
                     \pgfmathsetmacro{\x}{0.5}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \begin{center}
            %   \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 4.0 * x_n *(1 - x_n),\ \ x_0=0.501$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{4.0}
                     \pgfmathsetmacro{\x}{0.501}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
\pagedone
          \begin{center}
           %    \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,3) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 4.0 * x_n *(1 - x_n),\ \ x_0=0.75$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{4.0}
                     \pgfmathsetmacro{\x}{0.75}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \begin{center}
            %   \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,3) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 4.0 * x_n *(1 - x_n),\ \ x_0=0.751$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{4.0}
                     \pgfmathsetmacro{\x}{0.751}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
\pagedone
          \begin{center}
            %   \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,3) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 4.0 * x_n *(1 - x_n),\ \ x_0=0.751$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{4.0}
                     \pgfmathsetmacro{\x}{0.751}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7.4,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \begin{center}
            %   \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,3) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 4.0 * x_n *(1 - x_n),\ \ x_0=0.751$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{4.0}
                     \pgfmathsetmacro{\x}{0.751}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.99,-7.98,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
\pagedone
          \begin{center}
            %   \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,1) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 4.0 * x_n *(1 - x_n),\ \ x_0=0.3$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{4.0}
                     \pgfmathsetmacro{\x}{0.3}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.99,-7.98,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \begin{center}
            %   \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,1) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-1,-1) node{$x_{n+1} = 4.0 * x_n *(1 - x_n),\ \ x_0=0.01$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{4.0}
                     \pgfmathsetmacro{\x}{0.01}
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.99,-7.98,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
     \end{itemize}
\pagedone
\subsectionhead{Cobweb Diagrams}
     \begin{itemize}
          \item We can view the system in a somewhat different way.  This is a form of {\em phase space} for the system,
               where we will plot $x_{n+1}$ against $x_n$.  For any value of $R > 0$, the right hand side is a parabola opening down,
               with roots at $x_n = 0$ and $x_n = 1$.  The maximum value of the parabola occurs at $x_n = \frac{1}{2}$,
               and the maximum value is $\frac{R}{4}$.
               \begin{center}
                    \vspace{-0.5cm}
                \begin{tikzpicture}[scale=7, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.28, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\r}{3.0}
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
           %          \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
             %        \draw[color=purple](0.5,0.75) parabola[bend at start] (1,0);
                     \draw(1.58, 0.6) node {$x_{n+1}=Rx_n(1-x_n)$};
                     \draw [->] (1.4,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
\pagedone
               Now we'll follow the trajectory of the system.  We'll visualize this by drawing a sequence of lines (often called a
               cobweb diagram).  The coordinates of the endpoints of the lines will be
               $(x_0,0) -> (x_0, x_1),\ \ (x_0, x_1) -> (x_1, x_1),$\newline
               $(x_1, x_1) -> (x_1, x_2),\ \ (x_1, x_2) -> (x_2, x_2)$, etc.
               \begin{center}
                \begin{tikzpicture}[scale=7, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\r}{3.9}
                     \pgfmathsetmacro{\x}{0.4}
                     \pgfmathsetmacro{\wid}{1}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
          %           \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
            %         \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.65, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
                     \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw[line width=\wid,->](\x,0) -- (\x,\y/2);
                     \draw[line width=\wid,->](\x,\y/2) -- (\x,\y);
                     \pgfmathsetmacro{\midx}{\x+0.5*(\y-\x)}
                     \draw[line width=\wid,->](\x,\y) -- (\midx,\y);
                     \draw[line width=\wid,->](\midx,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,4} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \pgfmathsetmacro{\midy}{\y+0.5*(\x-\y)}
                          \draw[line width=\wid,->](\x,\x) -- (\x,\midy);
                          \draw[line width=\wid,->](\x,\midy) -- (\x,\y);
                          \pgfmathsetmacro{\midx}{\x+0.5*(\y-\x)}
                          \draw[line width=\wid,->](\x,\y) -- (\midx,\y);
                          \draw[line width=\wid,->](\midx,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
           Let's let that same example run for a while:
           \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.8}
                     \pgfmathsetmacro{\r}{3.9}
                     \pgfmathsetmacro{\x}{0.4}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0 = \x$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
             %        \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
               %      \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,30} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          And even longer:
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.8}
                     \pgfmathsetmacro{\r}{3.9}
                     \pgfmathsetmacro{\x}{0.4}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$ = \x};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
               %      \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                 %    \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,100} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          With a different starting value:
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.8}
                     \pgfmathsetmacro{\r}{3.9}
                     \pgfmathsetmacro{\x}{0.1}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$ = \x};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
               %      \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                 %    \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,100} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          And even longer:
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.9}
                     \pgfmathsetmacro{\x}{0.1}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$ = \x};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
             %        \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
               %      \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,350} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
\pagedone
          For other values of $R$ (320 iterations):
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.828}
                     \pgfmathsetmacro{\x}{0.5}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0 = \x$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
               %      \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                 %    \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,320} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          And slight changes (also 320 iterations):
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.829}
                     \pgfmathsetmacro{\x}{0.5}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0 = \x$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
               %      \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                 %    \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,320} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}

\pagedone
          \item Now we'll be a little more systematic in our analysis.  We'll start with small values of $R$, where
               the dynamics are relatively simple.  Let's look at the slope of the tangent line to the parabola
               at $0$.  We are working with the parabola $P(x) = R * x * (1 - x)$, or $P(x) = - Rx^2 + Rx$,
               so the derivative is $P'(x) = -2Rx + R$.  We thus have $P'(0) = R$, and hence when $R \le 1$,
               the entire parabola (for $x > 0$) lies below the line $x_{n+1} = x_n$.  In this case, the dynamics
               just die away to zero (i.e., $\lim_{n \to \infty}(x_n) = 0$).  In particular, $0$ is an {\em attracting
               fixed point} of the system.  (Note that for any value of $R$, $0$ is a fixed point of the system.)
               
               For $R \le 1$, the dynamics don't depend in any significant way on the starting value -- almost
               immediately, the system begins decaying to $0$, and continues directly there.
\pagedone
               \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{0.9}
                     \pgfmathsetmacro{\x}{0.7}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
               %      \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                 %    \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          For $R > 1$, we have a new feature -- the parabola crosses the line $x_{n+1} = x_n$ at some point
          $x > 0$.  In particular, we solve for that crossing point by setting $x_n = x_{n+1}$ -- in other words,
          we have
               $$ x_n = R * x_n * (1 - x_n),$$
          and so
          \begin{align*}
               x_n - Rx_n(1-x_n) &= 0 \\
               x_n - Rx_n + Rx_n^2 &= 0 \\
               Rx_n^2 + (1 - R)x_n &= 0 \\
               x_n(Rx_n + 1 - R) &= 0,
          \end{align*}
          and hence the crossing point occurs at
          $$x_n = \frac{R - 1}{R}.$$
\pagedone
          For all values of $R > 1$, this crossing point is a fixed point of the system.  For values $1 < R < 3$, this is
          an attracting fixed point for any starting value $x_0$ (except $x_0 = 0$ or $1$, and $0$ is now a {\em repelling}
          fixed point):
          \begin{center}
                \begin{tikzpicture}[scale=9, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{1.5}
                     \pgfmathsetmacro{\x}{0.2}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
               %      \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                 %    \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
 \pagedone
           \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{1.5}
                     \pgfmathsetmacro{\x}{0.9}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
              %       \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                %     \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
           \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{2.0}
                     \pgfmathsetmacro{\x}{0.9}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
             %        \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
               %      \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
           \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{2.0}
                     \pgfmathsetmacro{\x}{0.05}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
             %        \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
               %      \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
           \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{2.8}
                     \pgfmathsetmacro{\x}{0.9}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                %     \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                  %   \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
           Note that for $2 < R < 3$, the parabola is coming back down at the fixed point $\frac{R - 1}{R}$, and so the
           trajectory spirals in \ldots
\pagedone
     When $R = 3$, we have a new phenomenon.  We still have a fixed point at $\frac{R - 1}{R}$, but it is no longer
     and attracting fixed point:
     \begin{center}
                \begin{tikzpicture}[scale=9, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.0}
                     \pgfmathsetmacro{\x}{0.9}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                %     \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                  %   \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.0}
                     \pgfmathsetmacro{\x}{0.63}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                %     \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                  %   \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,50} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.0}
                     \pgfmathsetmacro{\x}{0.5}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
              %       \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                %     \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,50} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          For $R$ slightly bigger than $3$, we have new behavior:
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.075}
                     \pgfmathsetmacro{\x}{0.5}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                 %    \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                   %  \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          Or, starting closer to the fixed point:
          \begin{center}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.075}
                     \pgfmathsetmacro{\x}{0.65}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
               %      \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                 %    \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,40} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}

\pagedone
          As $R$ passes through $3$, the fixed point goes from being an attracting fixed to point to being
          an {\em unstable fixed point} (note that we actually already observed this as $R$ passed through
          $1$, where $0$ became an unstable fixed point).
          
          By looking more closely at the region of the fixed point as $R$ passes through $3$, we can
          understand better what is happening.  In particular, the critical feature of the system is that
          as $R$ goes through $3$, the slope of the tangent to the parabola goes from above $-1$ to
          below $-1$ (i.e., becomes more steep).
\pagedone
          \begin{center}
                \begin{tikzpicture}[scale=85, line width=1.75pt]
                     \draw(0.63,0.63) -- (0.69,0.63);
                     \draw(0.69,0.63) -- (0.69,0.69);
                     \draw(0.69,0.69) -- (0.63,0.69);
                     \draw(0.63,0.69) -- (0.63,0.63);
                     \draw(0.63,0.63) -- (0.69, 0.69);
                     \draw(0.62,0.63) node {$0.63$};
                     \draw(0.62,0.69) node {$0.69$};
                     \draw(0.69,0.625) node {$0.69$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{2.95}
                     \pgfmathsetmacro{\x}{0.64}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw[color=purple,domain=0.63:0.69,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)}) 
	                  coordinate(end);
                     \draw(0.75, 0.66) node {$x_{n+1}=\r*x_n(1-x_n)$};
                     \draw(0.653,0.62) node {$x_0 = \x$};
                     \draw[->](\x,0.623) -- (\x,0.63);
                     \draw[line width=\wid](\x,0.63) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,40} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
           \begin{center}
                \begin{tikzpicture}[scale=85, line width=1.75pt]
                     \draw(0.64,0.64) -- (0.69,0.64);
                     \draw(0.69,0.64) -- (0.69,0.69);
                     \draw(0.69,0.69) -- (0.64,0.69);
                     \draw(0.64,0.69) -- (0.64,0.64);
                     \draw(0.64,0.64) -- (0.69, 0.69);
                     \draw(0.63,0.64) node {$0.64$};
                     \draw(0.63,0.69) node {$0.69$};
                     \draw(0.69,0.635) node {$0.69$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.0}
                     \pgfmathsetmacro{\x}{0.645}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw[color=purple,domain=0.641:0.69,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)}) 
	                  coordinate(end);
                     \draw(0.75, 0.66) node {$x_{n+1}=\r*x_n(1-x_n)$};
                     \draw(0.663,0.62) node {$x_0 = \x$};
                     \draw[->](\x,0.623) -- (\x,0.64);
                     \draw[line width=\wid](\x,0.64) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,10} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
            \begin{center}
                \begin{tikzpicture}[scale=85, line width=1.75pt]
                     \draw(0.64,0.64) -- (0.69,0.64);
                     \draw(0.69,0.64) -- (0.69,0.69);
                     \draw(0.69,0.69) -- (0.64,0.69);
                     \draw(0.64,0.69) -- (0.64,0.64);
                     \draw(0.64,0.64) -- (0.69, 0.69);
                     \draw(0.63,0.64) node {$0.64$};
                     \draw(0.63,0.69) node {$0.69$};
                     \draw(0.69,0.635) node {$0.69$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.0}
                     \pgfmathsetmacro{\x}{0.65}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw[color=purple,domain=0.641:0.69,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)}) 
	                  coordinate(end);
                     \draw(0.75, 0.66) node {$x_{n+1}=\r*x_n(1-x_n)$};
                     \draw(0.663,0.62) node {$x_0 = \x$};
                     \draw[->](\x,0.623) -- (\x,0.64);
                     \draw[line width=\wid](\x,0.64) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,10} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
           \begin{center}
                \begin{tikzpicture}[scale=85, line width=1.75pt]
                     \draw(0.64,0.64) -- (0.69,0.64);
                     \draw(0.69,0.64) -- (0.69,0.69);
                     \draw(0.69,0.69) -- (0.64,0.69);
                     \draw(0.64,0.69) -- (0.64,0.64);
                     \draw(0.64,0.64) -- (0.69, 0.69);
                     \draw(0.63,0.64) node {$0.64$};
                     \draw(0.63,0.69) node {$0.69$};
                     \draw(0.69,0.635) node {$0.69$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.0}
                     \pgfmathsetmacro{\x}{0.655}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw[color=purple,domain=0.641:0.69,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)}) 
	                  coordinate(end);
                     \draw(0.75, 0.66) node {$x_{n+1}=\r*x_n(1-x_n)$};
                     \draw(0.663,0.62) node {$x_0 = \x$};
                     \draw[->](\x,0.623) -- (\x,0.64);
                     \draw[line width=\wid](\x,0.64) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,10} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
           \begin{center}
                \begin{tikzpicture}[scale=110, line width=1.75pt]
                     \draw(0.63,0.63) -- (0.71,0.63);
                     \draw(0.71,0.63) -- (0.71,0.71);
                     \draw(0.71,0.71) -- (0.63,0.71);
                     \draw(0.63,0.71) -- (0.63,0.63);
                     \draw(0.63,0.63) -- (0.71, 0.71);
                     \draw(0.62,0.63) node {$0.63$};
                     \draw(0.62,0.71) node {$0.71$};
                     \draw(0.71,0.625) node {$0.71$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.05}
                     \pgfmathsetmacro{\x}{0.667}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw[color=purple,domain=0.6303:0.708,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)}) 
	                  coordinate(end);
                     \draw(0.77, 0.66) node {$x_{n+1}=\r*x_n(1-x_n)$};
                     \draw(0.678,0.615) node {$x_0 = \x$};
                     \draw[->](\x,0.62) -- (\x,0.63);
                     \draw[line width=\wid](\x,0.63) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,40} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
           If we zoom out from this, we see another interesting phenomenon arise \ldots
           \begin{center}
                \begin{tikzpicture}[scale=55, line width=1.75pt]
                     \draw(0.57,0.57) -- (0.77,0.57);
                     \draw(0.77,0.57) -- (0.77,0.77);
                     \draw(0.77,0.77) -- (0.57,0.77);
                     \draw(0.57,0.77) -- (0.57,0.57);
                     \draw(0.57,0.57) -- (0.77, 0.77);
                     \draw(0.55,0.57) node {$0.57$};
                     \draw(0.55,0.77) node {$0.77$};
                     \draw(0.77,0.562) node {$0.77$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.05}
                     \pgfmathsetmacro{\x}{0.667}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw[color=purple,domain=0.5703:0.752,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)}) 
	                  coordinate(end);
                     \draw(0.86, 0.66) node {$x_{n+1}=\r*x_n(1-x_n)$};
                     \draw(0.678,0.555) node {$x_0 = \x$};
                     \draw[->](\x,0.56) -- (\x,0.57);
                     \draw[line width=\wid](\x,0.57) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,120} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
           \begin{center}
                \begin{tikzpicture}[scale=55, line width=1.75pt]
                     \draw(0.57,0.57) -- (0.77,0.57);
                     \draw(0.77,0.57) -- (0.77,0.77);
                     \draw(0.77,0.77) -- (0.57,0.77);
                     \draw(0.57,0.77) -- (0.57,0.57);
                     \draw(0.57,0.57) -- (0.77, 0.77);
                     \draw(0.55,0.57) node {$0.57$};
                     \draw(0.55,0.77) node {$0.77$};
                     \draw(0.77,0.562) node {$0.77$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.05}
                     \pgfmathsetmacro{\x}{0.575}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw[color=purple,domain=0.5703:0.752,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)}) 
	                  coordinate(end);
                     \draw(0.86, 0.66) node {$x_{n+1}=\r*x_n(1-x_n)$};
                     \draw(0.61,0.555) node {$x_0 = \x$};
                     \draw[->](\x,0.56) -- (\x,0.57);
                     \draw[line width=\wid](\x,0.57) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,120} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
\pagedone
          \item We are seeing here an orbit of period 2 -- the system bounces back and forth between two values.
               For values of $R < 3$ we had an attracting fixed point.  The attracting fixed point is at $0$ for $0 \le R \le 1$,
               and at $\frac{R-1}{R}$ for $1 < R < 3$.  At $R = 3$ we still have the fixed point (and in fact that fixed
               point remains for all $R \ge 1$), but it is no longer a stable fixed point.  For $R > 3$, it is a {\em repelling
               fixed point} -- values near, but not exactly on, the fixed point will, in successive iterations, move
               further away.
               
               At $R = 1$ and $R = 3$, changes in $R$ result in significant changes in the dynamics of the system.
               These significant changes in the dynamics of the system resulting from changes in a controlling
               parameter are called {\em bifurcations}.  At $R = 1$, the stable (attracting) fixed point
               at $0$ becomes unstable, and becomes a repelling fixed point.  We acquire a new stable (attracting)
               fixed point at $\frac{R-1}{R}$.
\pagedone
               At $R = 3$, the fixed point at $\frac{R-1}{R}$ becomes unstable, and we acquire a new attracting stable
               orbit of period 2.  These is the first of a sequence of bifurcations in the dynamics of the logistics
               equation.  At each of these bifurcations, an orbit of period $2^n$ becomes unstable (although it
               continues to exist in the dynamics), and a new stable orbit of period $2^{n+1}$ arises.  This process
               is called a {\em period doubling bifurcation cascade}.
               
               We can calculate the $x$ values of this period 2 orbit by looking at two steps of the difference equation:
               \begin{align*}
                    x_{n+2} &= Rx_{n+1}(1 - x_{n+1}) \\
                                  &= R(Rx_n(1 - x_n))(1 - Rx_n(1 - x_n)) \\
                                  &= R^2x_n(1 - x_n)(1 - Rx_n + Rx_n^2) \\
                                  &= R^2x_n(-Rx_n^3 + 2Rx_n^2 - (R + 1)x_n + 1)
               \end{align*}
\pagedone
               We are looking for period 2 orbits -- in other words, values where $x_{n+2} = x_n$.  We therefore want
                    $$x_n = R^2x_n(-Rx_n^3 + 2Rx_n^2 - (R + 1)x_n + 1)$$
               or
                    $$R^2x_n(-Rx_n^3 + 2Rx_n^2 - (R + 1)x_n + 1) - x_n = 0.$$
               This is a 4th degree polynomial is $x_n$, which, for $R > 3$, has 4 roots.  It is easy to see that
               $x_n = 0$ is a root (but we already knew that, because 0 is a fixed point of the original equation,
               and hence also repeats itself every 2 steps).  Similarly, $x_n = \frac{R-1}{R}$ is a root, because
               it too is a fixed point.  We could proceed with the (somewhat messy) algebra to find the other
               two roots, but instead let's look at the cobweb diagram for $x_{n+2}$ vs. $x_n$ (in these diagrams,
               we're also seeing the original parabolas, for reference \ldots).
\pagedone
               You can see the bifurcation happen as $R$ reaches 3.0:
               \begin{center}
                    \vspace{-0.5cm}
                \begin{tikzpicture}[scale=5.5, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=green](-0.2, 0.6) node {$x_{n+2}$};
                     \draw[color=purple](-0.2, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.4, 0.95) node {$x_{n+2}=x_n$};
                     \pgfmathsetmacro{\r}{2.8}
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                     \draw[color=green,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
	                  coordinate(end);
                     \draw(1.5, 0.6) node {$R = \r$};
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
           \begin{center}
                \vspace{-0.5cm}
                \begin{tikzpicture}[scale=5.5, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=green](-0.2, 0.6) node {$x_{n+2}$};
                     \draw[color=purple](-0.2, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.4, 0.95) node {$x_{n+2}=x_n$};
                     \pgfmathsetmacro{\r}{3.0}
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                     \draw[color=green,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
	                  coordinate(end);
                     \draw(1.5, 0.6) node {$R = \r$};
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
 \pagedone
           and moves on past \ldots
           \begin{center}
                \vspace{-0.5cm}
                \begin{tikzpicture}[scale=5.5, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=green](-0.2, 0.6) node {$x_{n+2}$};
                     \draw[color=purple](-0.2, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.4, 0.95) node {$x_{n+2}=x_n$};
                     \pgfmathsetmacro{\r}{3.2}
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                     \draw[color=green,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
	                  coordinate(end);
                     \draw(1.5, 0.6) node {$R = \r$};
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
           \begin{center}
                \vspace{-0.5cm}
                \begin{tikzpicture}[scale=5.5, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=green](-0.2, 0.6) node {$x_{n+2}$};
                     \draw[color=purple](-0.2, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.4, 0.95) node {$x_{n+2}=x_n$};
                     \pgfmathsetmacro{\r}{3.7}
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                     \draw[color=green,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
	                  coordinate(end);
                     \draw(1.5, 0.6) node {$R = \r$};
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
\pagedone
               We can also look at higher order iterates:
               \begin{center}
               % \vspace{-0.5cm}
                \begin{tikzpicture}[scale=12, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=blue](-0.15, 0.7) node {$x_{n+4}$};
                     \draw[color=green](-0.15, 0.5) node {$x_{n+2}$};
                     \draw[color=purple](-0.15, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw[color=blue](1.23, 1) node {$x_{n+4}=x_n$};
                     \pgfmathsetmacro{\r}{3.829}
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                     \draw[color=green,domain=0:1,
	                  smooth,samples=600,variable=\t]  plot
	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
	                  coordinate(end);
	             \draw[color=blue,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1))*(1-(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)))})
	                  coordinate(end);
                     \draw(1.2, 0.6) node {$R = \r$};
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
\pagedone
               For various values of $R$:
               \begin{center}
                \vspace{0.07cm}
                \begin{tikzpicture}[scale=12, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=blue](-0.15, 0.7) node {$x_{n+4}$};
                     \draw[color=green](-0.15, 0.5) node {$x_{n+2}$};
                     \draw[color=purple](-0.15, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw[color=blue](1.23, 1) node {$x_{n+4}=x_n$};
                     \pgfmathsetmacro{\r}{3.9}
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
                     \draw[color=green,domain=0:1,
	                  smooth,samples=600,variable=\t]  plot
	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
	                  coordinate(end);
	             \draw[color=blue,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1))*(1-(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)))})
	                  coordinate(end);
                     \draw(1.23, 0.6) node {$R = \r$};
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
\pagedone
          We can do cobweb diagrams also (320 iterations):
          \begin{center}
               % \vspace{-0.5cm}
                \begin{tikzpicture}[scale=11, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=blue](-0.15, 0.7) node {$x_{n+4}$};
           %          \draw[color=green](-0.15, 0.5) node {$x_{n+2}$};
              %       \draw[color=purple](-0.15, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw[color=blue](1.23, 1) node {$x_{n+4}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.829}
                     \pgfmathsetmacro{\x}{0.3}
%                     \draw[color=purple,domain=0:1,
%	                  smooth,samples=600,variable=\t] plot
%	                  ({\t},{\r*\t*(1-\t)})
%	                  coordinate(end);
%                     \draw[color=green,domain=0:1,
%	                  smooth,samples=600,variable=\t]  plot
%	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
%	                  coordinate(end);
	             \draw[color=blue,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1))*(1-(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)))})
	                  coordinate(end);
                     \draw(1.3, 0.6) node {$R = \r$};
                     \draw(\x, -0.06) node{$x_0 = \x$};
                  %    \draw(1.268, 0.4) node {$x_0 = \x$};
                     \pgfmathsetmacro{\y}{\r*(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1))*(1-(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1)))}
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,320} {
                          \xdef\x{\y}
                    %      \pgfmathparse{\r*\x*(1-\x)}
                          \pgfmathparse{\r*(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1))*(1-(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1)))}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
\pagedone
          With changing starting values (still 320 iterations):
          \begin{center}
               % \vspace{-0.5cm}
                \begin{tikzpicture}[scale=11, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=blue](-0.15, 0.7) node {$x_{n+4}$};
           %          \draw[color=green](-0.15, 0.5) node {$x_{n+2}$};
              %       \draw[color=purple](-0.15, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw[color=blue](1.23, 1) node {$x_{n+4}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.829}
                     \pgfmathsetmacro{\x}{0.5}
%                     \draw[color=purple,domain=0:1,
%	                  smooth,samples=600,variable=\t] plot
%	                  ({\t},{\r*\t*(1-\t)})
%	                  coordinate(end);
%                     \draw[color=green,domain=0:1,
%	                  smooth,samples=600,variable=\t]  plot
%	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
%	                  coordinate(end);
	             \draw[color=blue,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1))*(1-(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)))})
	                  coordinate(end);
                     \draw(1.3, 0.6) node {$R = \r$};
                     \draw(\x, -0.06) node{$x_0 = \x$};
                  %    \draw(1.268, 0.4) node {$x_0 = \x$};
                     \pgfmathsetmacro{\y}{\r*(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1))*(1-(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1)))}
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,320} {
                          \xdef\x{\y}
                    %      \pgfmathparse{\r*\x*(1-\x)}
                          \pgfmathparse{\r*(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1))*(1-(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1)))}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
\pagedone
          And changing values of $R$ (also 320 iterations):
          \begin{center}
               % \vspace{-0.5cm}
                \begin{tikzpicture}[scale=11, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=blue](-0.15, 0.7) node {$x_{n+4}$};
           %          \draw[color=green](-0.15, 0.5) node {$x_{n+2}$};
              %       \draw[color=purple](-0.15, 0.3) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw[color=blue](1.23, 1) node {$x_{n+4}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.828}
                     \pgfmathsetmacro{\x}{0.5}
%                     \draw[color=purple,domain=0:1,
%	                  smooth,samples=600,variable=\t] plot
%	                  ({\t},{\r*\t*(1-\t)})
%	                  coordinate(end);
%                     \draw[color=green,domain=0:1,
%	                  smooth,samples=600,variable=\t]  plot
%	                  ({\t},{pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)})
%	                  coordinate(end);
	             \draw[color=blue,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1))*(1-(pow(\r,2)*\t*((-1)*\r*pow(\t,3)+2*\r*pow(\t,2)-(\r+1)*\t+1)))})
	                  coordinate(end);
                     \draw(1.3, 0.6) node {$R = \r$};
                     \draw(\x, -0.06) node{$x_0 = \x$};
                  %    \draw(1.268, 0.4) node {$x_0 = \x$};
                     \pgfmathsetmacro{\y}{\r*(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1))*(1-(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1)))}
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,320} {
                          \xdef\x{\y}
                    %      \pgfmathparse{\r*\x*(1-\x)}
                          \pgfmathparse{\r*(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1))*(1-(pow(\r,2)*\x*((-1)*\r*pow(\x,3)+2*\r*pow(\x,2)-(\r+1)*\x+1)))}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
     \end{itemize}
\pagedone
\subsectionhead{Bifurcation Diagrams}
     \begin{itemize}
          \item There is another way of viewing the behavior of the logistics equation, which makes it easier
               to see the changes in behavior as $R$ changes.  We will display trajectories for various values
               of $R$ all at once in a single two dimensional picture.  On our two dimensional image, the
               horizontal axis will be for various values of $R$.  For each value of $R$, the vertical
               direction will show a sequence of values of $x_n$.
               
               This approach will allow us to see overall changes in the dynamics as we vary the
               controlling parameter $R$.  As we will see, there are values of $R$ at which we
               will have significant changes in the dynamics.  As indicated above, these changes
               are called {\em bifurcations}, and hence this view is often called a {\em bifurcation
               diagram}.
\pagedone
               Thus, for example, when $R = 3.5$ (and $x_0 = 0.5$):
                \begin{center}
               \vspace{-0.5cm}
               \begin{tikzpicture}[line width=1.75pt]
                     \draw[color=black, ->] (-8.2,0) -- (8,0);
                     \draw[color=black](-8.6,2) node{$x_n$};
                     \draw[color=black, ->](-8,-0.2) -- (-8,4.4);
                     \draw[color=black](-8.6,4) node{$1$};
                     \draw[color=black](-8.2,4) -- (-7.8,4);
                     \draw[color=black](8.4,0) node{$n$};
                     \pgfmathsetmacro{\R}{3.5}
                     \pgfmathsetmacro{\x}{0.3}
                      \draw[color=black](-1,-1) node{$x_{n+1} = \R * x_n *(1 - x_n),\ \ x_0=0.5$};
                     \draw[fill=black](-8,4*\x) circle(0.05cm);
                     \foreach \i in {-7.5,-7,...,7} {
                          \pgfmathparse{\R*\x*(1-\x)}
                          \xdef\x{\pgfmathresult}
                          \draw[fill=black](\i,4*\x) circle(0.05cm);
                     }
               \end{tikzpicture}
          \end{center}
          \vspace{-0.75cm}
          We rotate the graph, and show values of $x_n$  at $R = 3.5$:
           \begin{center}
               \vspace{-0.5cm}
                \begin{tikzpicture}[scale=4, line width=1.75pt]
                     \draw[->](0,0) -- (4.1,0);
                     \draw(4.2,0) node {$R$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=black](-0.15, 0.5) node {$x_{n}$};
                     \draw[thin,dashed](0,1) -- (4,1);
                     \draw[thin,dashed](4,0) -- (4,1);
                     \draw(1,-0.05) -- (1,0.05);
                     \draw(2,-0.05) -- (2,0.05);
                     \draw(3,-0.05) -- (3,0.05);
                     \draw(4,-0.05) -- (4,0.05);
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.5}
                     \pgfmathsetmacro{\x}{0.3}
                     \draw(\r, -0.2) node {$R = \r$};
                     \draw(\r,-0.05) -- (\r,0.05);
                     \draw[fill=black](\r,\x) circle(0.01cm);
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \foreach \i in {1,2,...,20} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[fill=black](\r,\x) circle(0.01cm);
                     }
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
           This image is a bit of a smear, and it is hard to see structure in the dynamics.
           We are actually more interested in the long term behavior of the system.  For
           any particular starting value of $x_0$, the system is likely to wander around
           for a while before settling down to its stable (limiting) behavior.  We can
           see this for particular values of $R$.
\pagedone
          Here we are looking at $R = 3.5$.  The system wanders for a while, but
          is settling down to an orbit of period 4:
           \begin{center}
                \vspace{-0.5cm}
                \begin{tikzpicture}[scale=10, line width=1.75pt]
                     \draw[->](0,0) -- (1.1,0);
                     \draw(1.2,0) node {$x_n$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.06, 1) node {$1$};
                     \draw(-0.2, 0.4) node {$x_{n+1}$};
                     \draw[->](0,0) -- (1.05,1.05);
                     \draw[thin,dashed](0,1) -- (1,1);
                     \draw[thin,dashed](1,0) -- (1,1);
                     \draw(1.29, 0.95) node {$x_{n+1}=x_n$};
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.5}
                     \pgfmathsetmacro{\x}{0.3}
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \draw(\x, -0.06) node{$x_0 = \x$};
                     \draw(1, -0.06) node{$1$};
                     \draw[color=purple,domain=0:1,
	                  smooth,samples=600,variable=\t] plot
	                  ({\t},{\r*\t*(1-\t)})
	                  coordinate(end);
               %      \draw[color=purple](0,0) parabola[bend at end] (0.5,\r/4);
                 %    \draw[color=purple](0.5,\r/4) parabola[bend at start] (1,0);
                     \draw(1.5, 0.5) node {$x_{n+1}=\r*x_n(1-x_n)$};
               %      \draw [->] (1.65,0.45) arc (-45:-89:1.0);
                     \draw(1.328, 0.33) node {$x_0 = \x$};
                     \draw[line width=\wid](\x,0) -- (\x,\y);
                     \draw[line width=\wid](\x,\y) -- (\y,\y);
                     \foreach \i in {1,2,...,40} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[line width=\wid](\x,\x) -- (\x,\y);
                          \draw[line width=\wid](\x,\y) -- (\y,\y);
                     }
                \end{tikzpicture}
           \end{center}
 \pagedone
           What we will do is to throw away early values of $x_n$ (often called the {\em transients}),
           and show the long term, or {\em limiting} trajectory.  In this picture, we have thrown away
           the first 80 values of $x_n$, and plotted the next 40:
           \begin{center}
               % \vspace{-0.5cm}
                \begin{tikzpicture}[scale=4, line width=1.75pt]
                     \draw[->](0,0) -- (4.1,0);
                     \draw(4.2,0) node {$R$};
                     \draw[->](0,0) -- (0,1.05);
                     \draw(-0.02,1) -- (0.02,1);
                     \draw(-0.1, 1) node {$1$};
                     \draw[color=black](-0.15, 0.5) node {$x_{n}$};
                     \draw[thin,dashed](0,1) -- (4,1);
                     \draw[thin,dashed](4,0) -- (4,1);
                     \draw(1,-0.05) -- (1,0.05);
                     \draw(2,-0.05) -- (2,0.05);
                     \draw(3,-0.05) -- (3,0.05);
                     \draw(4,-0.05) -- (4,0.05);
                     \pgfmathsetmacro{\wid}{0.6}
                     \pgfmathsetmacro{\r}{3.5}
                     \pgfmathsetmacro{\x}{0.3}
                     \draw(\r, -0.2) node {$R = \r$};
                     \draw(\r,-0.05) -- (\r,0.05);
                   %  \draw[fill=black](\r,\x) circle(0.01cm);
                     \pgfmathsetmacro{\y}{\r*\x*(1-\x)}
                     \foreach \i in {1,2,...,80} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                        %  \draw[fill=black](\r,\x) circle(0.01cm);
                     }
                     \foreach \i in {1,2,...,40} {
                          \xdef\x{\y}
                          \pgfmathparse{\r*\x*(1-\x)}
                          \xdef\y{\pgfmathresult}
                          \draw[fill=black](\r,\x) circle(0.01cm);
                     }
                  %   \draw [->] (1.46,0.55) arc (-45:-90:0.7);
                \end{tikzpicture}
           \end{center}
           In this picture, we can see the period 4 orbit, which is an attracting closed
           periodic orbit.  As $n$ goes to $\infty$, trajectories approach this limiting
           orbit.
\pagedone
               In the following images, we can see "bifurcation diagrams" for the logistic equation.  In
               the first couple of images, things are monochrome, with dots for values of $x_n$.
               In the later pictures, there are various colors, reflecting the amount of time the system
               spends in those regions (closer to white means longer time spent).
               
               The first diagram shows $0 \le R \le 4$, and $0 \le x_n \le 1$.  Later we can talk about why
               we keep $R \le 4$.  These images all have $x_0 = 0.3$, and discard the first 80 iterations.
               
               In the first image, you can see the transition from a stable (attracting) fixed point at 0 for
               $R < 1$.  At $R = 1$, that fixed point becomes unstable, and there is a new stable
               (attracting) fixed point at $\frac{R-1}{R}$ for $1 < R < 3$.  At $R = 3$ that fixed point
               becomes unstable, and a new attractor of period 2 arises.  As we increase $R$, that
               period 2 orbit becomes unstable, and we get a new orbit of period 4, then period 8, etc.
               As $R$ grows, we pass through orbits of all periods of the form $2^n$.  This is often
               referred to as a {\em period doubling bifurcation cascade}.  This period doubling cascade
               reaches its limit before $R$ goes to $\infty$, and we enter a chaotic realm.
               
               Other diagrams show some of what we will see if we zoom in.  The captions under
               the images tell the range of $R$ values, the range of values for $x_n$ being shown,
               and the number of iterations.  In these images, there are 900 possible values for $x_n$.
               In generating the images, the logistics equation is iterated for a given value of $R$, and
               at each step one of 900 bins for $x_n$ is incremented.  These values are then used to
               select a color.
               
               It is worth remembering that these images, as we zoom in, also reveal various artifacts
               having to do with machine resolution, etc. \ldots
               
               \centerline{\includegraphics[width = 7.5in]{snap-0_000-4_000-0_000-1_000-1500.jpg}}
               \centerline{R=0.000-R=4.000-x=0.000-x=1.000-iter=1500}
               \centerline{\includegraphics[width = 7.5in]{snap-2_800-4_000-0_000-1_000-1500.jpg}}
               \centerline{R=2.800-R=4.000-x=0.000-x=1.000-iter=1500}
               
               \centerline{\includegraphics[width = 7.5in]{snap-2_800-4_000-0_000-1_000-80000.jpg}}
               \centerline{R=2.800-R=4.000-x=0.000-x=1.000-iter=80000}
               \centerline{\includegraphics[width = 7.5in]{snap-3_823-3_869-0_447-0_564-80000-z.jpg}}
               \centerline{R=2.8000-R=4.000-x=0.000-x=1.000-iter=80000-z}
                \centerline{\includegraphics[width = 7.5in]{snap-3_823-3_869-0_447-0_564-320000.jpg}}
                \centerline{R=3.823-R=3.869-x=0.447-x=0.564-iter=320000}
               \centerline{\includegraphics[width = 7.5in]{snap-2_997-3_002-0_660-0_677-80000-z.jpg}}
               \centerline{R=2.8000-R=4.000-x=0.000-x=1.000-iter=80000}
               \centerline{\includegraphics[width = 7.5in]{snap-2_997-3_002-0_660-0_677-80000.jpg}}
               \centerline{R=2.997-R=3.002-x=0.660-x=0.677-iter=80000}
               \centerline{\includegraphics[width = 7.5in]{snap-3_000-3_000-0_665-0_668-80000-z.jpg}}
               \centerline{R=3.000-R=3.000-x=0.665-x=0.668-iter=80000-z}
               \centerline{\includegraphics[width = 7.5in]{snap-3_000-3_000-0_665-0_668-160000.jpg}}
               \centerline{R=3.000-R=3.000-x=0.665-x=0.668-iter=160000}
               \centerline{\includegraphics[width = 7.5in]{snap-3_633-3_733-0_691-0_758-80000-z.jpg}}
               \centerline{R=2.800-R=4.000-x=0.000-x=1.000-iter=80000 z}
               \centerline{\includegraphics[width = 7.5in]{snap-3_633-3_733-0_691-0_758-320000.jpg}}
               \centerline{R=3.633-R=3.733-x=0.691-x=0.758-iter=320000}
               
  %             \centerline{\includegraphics[width = 7.5in]{snap-3_000-3_000-0_666-0_667-200000.jpg}}
    %           \centerline{R=3.000-R=3.000-x=0.666-x=0.667-iter=200000}
               
          \item There are various tools for exploring such bifurcation diagrams, such as:
          
                \hyperref{http://csustan.csustan.edu/~tom/SFI-CSSS/nonlinear/Chaos.html}{}{}
                     {http://csustan.csustan.edu/\~\ tom/SFI-CSSS/nonlinear/Chaos.html}
                     
      \end{itemize}
\pagedone
\subsectionhead{Universality}
     \begin{itemize}
          \item  An important result in the study of nonlinear (and chaotic) systems is a {\em universality}
               property discovered by Mitchell Feigenbaum.  As we increase $R$ from $0$, we see a
               sequence of bifurcations.  In particular, starting at $R = 3$, we see a sequence of period
               doubling bifurcations.  by the time we get to $R = 3.57$, we have had infinitely many
               period doublings, and we reach a chaotic realm.
               
               Feigenbaum observed that the distance between successive period doubling bifurcations
               in the logistics map exhibited an interesting property.  If we let $R_i$ be the location of the
               $i$th period doubling bifurcation after $R = 1$, and let $d_i = R_{i+1} - R_i$, then we
               can look at the sequence $\frac{d_{i}}{d_{i+1}}$.  Feigenbaum showed that this sequence
               has a limit:
                    $$\delta = \lim_{i \to \infty}\frac{d_{i}}{d_{i+1}}$$
\pagedone
              The value of $\delta$ has been calculated (to many decimal places):               
               $$\delta = 4.66920160910299067185320382\ldots$$
               This value is approached as $R$ approaches the accumulation point
               where the bifurcation cascade ends, and a chaotic region begins:
               $$R_{\infty} =3.569934669\ldots$$
               Feigenbaum also made a more striking and important observation.  This process
               (period doubling bifurcation cascade route to chaos) occurs in may systems,
               and the value of $\delta$ is {\em universal} for a broad range of systems.  This
               universality property opened up the possibility of finding {\em quantitative}
               properties, rather than just {\em qualitative} properties, for a variety of nonlinear
               systems.
               
\pagedone
               \centerline{\includegraphics[width = 7.5in]{snap-2_600-4_000-0_000-1_000-1500-delta.jpg}}
               \centerline{R=2.6-R=4.0-x=0.0-x=1.0-iter=1500 $\mathbf{d_i}$}  
\pagedone
              Feigenbaum also discovered another {\em universality} principle concerning the width
              of the tines in the pitchfork bifurcation diagrams.  In this case, we look at the
              successive widths of the tines, and find that the limiting value of the ratios of these
              widths is
              $$\alpha = 2.502907875095892822283902873218\ldots$$
              On the next page is a brief pictorial overview.
              
              For more discussion of these ideas, see, for example, Fiegenbaum \cite{feigenbaum}
              and Fillion \cite{fillion}.
                \centerline{\includegraphics[width = 7.5in]{snap-2_600-4_000-0_000-1_000-1500-alpha.jpg}}
               \centerline{R=2.6-R=4.0-x=0.0-x=1.0-iter=1500 $\mathbf{a_i}$} 
\pagedone
               An example of the range of universality of these properties can be seen from the following
               conditions, which are sufficient to give $\delta$ as the limiting value for the bifurcation
               cascade:
               \begin{enumerate}
                    \item $f: [0, 1] \to \myreal$ is continuous, with a unique differentiable maximum $\bar{x}$ ;
                    \item $f(0) = f(1) = 0$, $f(x) > 0$ for $x \in (0,1)$, f is strictly increasing on $(0,\bar{x})$, and
                            strictly decreasing on $(\bar{x}, 1)$;
                    \item For some parameter value, $f$ has two fixed points which are both unstable; and,
                    \item In the interval $N$ containing $\bar{x}$ such that $|f'(x)| < 1$, $f$ is concave downward.
               \end{enumerate}
      \end{itemize}              
\pagedone
 \sectionhead{The {\twntyfvcyb{Sharkovski\"i}} Theorem}
      \begin{itemize}
               \item  The {\twntyfrcyr{Sharkovski\"i}} Theorem tells us about {\em periodic points} of continuous functions.
                    (Note:  {\twntyfrcyr{Sharkovski\"i}} is a Ukrainian mathematician.  His name (here written in Cyrillic) is
                    transliterated as "Sarkovskii" or "Sharkovsky" or "Sharkovski" or \ldots  Since I can, I've gone ahead
                    and used the Cyrillic :-)
                    
                    Suppose$f(x)$  is a function.  Then a point $x$ is a {\em periodic point of period} $n$
                    for $f$ if
                         $$x = f^n(x) = f(f(\ldots(f(x))\ldots)),$$
                     and
                         $$x \ne f^k(x)\ \mathrm{for}\ 1 \le k < n.$$
\pagedone
                    The theorem looks at the special case of a continuous function $f: \myreal \to \myreal$.

                    Consider the following ordering of the positive natural numbers:
                    $$  \begin{array}{cccccccc}
			3 & 5 & 7 & 9 & 11 & \ldots & (2n+1)\cdot2^{0} & \ldots\\
			3\cdot2 & 5\cdot2 & 7\cdot2 & 9\cdot2 & 11\cdot2 & \ldots & (2n+1)\cdot2^{1} & \ldots\\
			3\cdot2^{2} & 5\cdot2^{2} & 7\cdot2^{2} & 9\cdot2^{2} & 11\cdot2^{2} & \ldots & (2n+1)\cdot2^{2} & \ldots\\
			3\cdot2^{3} & 5\cdot2^{3} & 7\cdot2^{3} & 9\cdot2^{3} & 11\cdot2^{3} & \ldots & (2n+1)\cdot2^{3} & \ldots\\
			 & \vdots\\
			\ldots & 2^{n} & \ldots & 2^{4} & 2^{3} & 2^{2} & 2 & 1\end{array}
		   $$
		   (note that every positive natural number occurs exactly once in this list).
               
               The {\twntyfrcyr{Sharkovski\"i}} Theorem states that if $f: \myreal \to \myreal$ is continuous, and has a
               periodic point $x_n$ of period $n$  for some value $n$ on this list, then it also has points $y_m$ of
               period $m$ for all $m$ later in
               the list than $n$.  In particular, if $f$ has a point $x_3$ of period 3, then it has periodic points of all
               possible periods.

               This leads to the famous saying that ``period 3 implies chaos'' (although this may need
               to be taken with a grain of salt \ldots).  There is a classic paper about this:  {\em Period Three Implies
               Chaos} by Li and Yorke \cite{li}.
               
     \end{itemize}
     
\pagedone
