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<math>\begin{array}{ll}

<math>\begin{array}{ll}

1.1. & f : \mathbb{B} \to \mathbb{B}\ \text{such that}\ f : \texttt{x} \mapsto \texttt{(x)}

1.1. & f : \mathbb{B} \to \mathbb{B} ~\text{such that}~ f : \texttt{x} \mapsto \texttt{(x)}

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1.2. & \texttt{x}' ~=~ \texttt{(x)}

1.2. & \texttt{x}' ~=~ \texttt{(x)}

{| align="center" cellpadding="8" width="90%"

{| align="center" cellpadding="8" width="90%"

| 2.1. || <math>F : \mathbb{B}^2 \to \mathbb{B}^2</math> such that <math>F : (u, v) \mapsto ( ~\underline{((}~ u ~\underline{)(}~ v ~\underline{))}~ , ~\underline{((}~ u ~,~ v ~\underline{))}~ )</math>

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| 2.2. || <math>u' ~=~ \underline{((}~ u ~\underline{)(}~ v ~\underline{))}~, \quad v' ~=~ ~\underline{((}~ u ~,~ v ~\underline{))}</math>

2.1. & F : \mathbb{B}^2 \to \mathbb{B}^2 ~\text{such that}~ F : (\texttt{u}, \texttt{v}) \mapsto ( ~\texttt{((u)(v))}~ , ~\texttt{((u,~v))}~ )

| 2.3. || <math>u ~:=~ \underline{((}~ u ~\underline{)(}~ v ~\underline{))}~, \quad v := ~\underline{((}~ u ~,~ v ~\underline{))}</math>

2.2. & \texttt{u}' ~=~ \texttt{((u)(v))}~, \quad \texttt{v}' ~=~ \texttt{((u,~v))}

| 2.4. || ???

2.3. & \texttt{u} ~:=~ \texttt{((u)(v))}~, \quad \texttt{v} ~:=~ \texttt{((u,~v))}

2.4. & ???

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What we are looking for is &mdash; one rule to rule them all, a rule that applies to every state and works every time.

What we are looking for is &mdash; one rule to rule them all, a rule that applies to every state and works every time.

What we see at first sight in the tables above are patterns of differential features that attach to the states in each orbit of the dynamics.  Looked at locally to these orbits, the isolated fixed point at <math>(1, 1)\!</math> is no problem, as the rule <math>du = dv = 0\!</math> describes it pithily enough.  When it comes to the other orbit, the first thing that comes to mind is to write out the law <math>du = v,\ dv = ~\underline{(}~ u ~\underline{)}~.</math>

What we see at first sight in the tables above are patterns of differential features that attach to the states in each orbit of the dynamics.  Looked at locally to these orbits, the isolated fixed point at <math>(1, 1)\!</math> is no problem, as the rule <math>\texttt{du~=~dv~=~0}</math> describes it pithily enough.  When it comes to the other orbit, the first thing that comes to mind is to write out the law <math>\texttt{du~=~v}, ~\texttt{dv~=~(u)}.</math>

==Note 13==

==Note 13==