Consider the Poincare maps defined in \(\left( {\bf{3}} \right)\) with \({\bf{\omega = }}\sqrt {\bf{2}} {\bf{,A = F = 1,}}\) and\(\phi = 0\). If this map were ever to repeat, then for two distinct positive integers \({\bf{n}}\)and\({\bf{m}}\), \({\bf{sin(2}}\sqrt {\bf{2}} {\bf{\pi n) = sin(2}}\sqrt {\bf{2}} {\bf{\pi m)}}{\bf{.}}\)Using basic properties of the sine function, show that this would imply that \(\sqrt {\bf{2}} \) is rational. It follows from this contradiction that the points of the Poincare map do not repeat.

For \({\bf{A = F = 1,\omega = }}\sqrt {\bf{2}} \) and \(\phi = 0\)the Poincare map is given by:

\[{{\bf{x}}_{\bf{n}}}{\bf{ = sin(2}}\sqrt {\bf{2}} {\bf{n\pi ) + 1 = sin(2}}\sqrt {\bf{2}} {\bf{m\pi ) + 1 = }}{{\bf{x}}_{\bf{m}}}\]

\[{{\bf{v}}_{\bf{n}}}{\bf{ = cos(2}}\sqrt {\bf{2}} {\bf{n\pi ) = cos(2}}\sqrt {\bf{2}} {\bf{n\pi ) = }}{{\bf{v}}_{\bf{m}}}\]

If this map repeats, one has that there exist some integers\({\bf{n}}\), \({\bf{m}}\) such that \({{\bf{x}}_{\bf{n}}}{\bf{ = }}{{\bf{x}}_{\bf{m}}}{\bf{, }}{{\bf{v}}_{\bf{n}}}{\bf{ = }}{{\bf{v}}_{\bf{m}}}{\bf{,}}\)i.e., such that:

\(\begin{aligned}{c}{\bf{sin(2}}\sqrt {\bf{2}} {\bf{n\pi ) + 1 = sin(2}}\sqrt {\bf{2}} {\bf{m\pi ) + 1}}\;\;\;{\bf{ and }}\;\;\;\sqrt {\bf{2}} {\bf{cos(2}}\sqrt {\bf{2}} {\bf{n\pi ) = }}\sqrt {\bf{2}} {\bf{cos(2}}\sqrt {\bf{2}} {\bf{m\pi ) }}\\{\bf{sin(2}}\sqrt {\bf{2}} {\bf{n\pi ) = sin(2}}\sqrt {\bf{2}} {\bf{m\pi ),}}\;\;\;{\bf{ and }}\;\;\;{\bf{cos(2}}\sqrt {\bf{2}} {\bf{n\pi ) = cos(2}}\sqrt {\bf{2}} {\bf{m\pi )}}{\bf{.}}\end{aligned}\)

One knows that \({\bf{sinx = siny}}\) and \({\bf{cosx = cosy}}\) if and only if \({\bf{x = y + 2k\pi }}\)for some \({\bf{k}} \in {\bf{Z,}}\)

So, one can now conclude that:

\({\bf{2}}\sqrt {\bf{2}} {\bf{n\pi = 2}}\sqrt {\bf{2}} {\bf{m\pi + 2k\pi ,}}\)for some \({\bf{k}} \in {\bf{Z,}}\)

Thus, one has:

\(\begin{aligned}{c}\sqrt {\bf{2}} {\bf{n = }}\sqrt {\bf{2}} {\bf{m + k}}\\\sqrt {\bf{2}} {\bf{(n - m) = k}}\;\\\sqrt {\bf{2}} {\bf{ = }}\frac{{\bf{k}}}{{{\bf{n - m}}}}\end{aligned}\)

One has that \({\bf{k}} \in {\bf{Z}}\)and \({\bf{n - m}} \in {\bf{Z,}}\)so the previous equality gives us that \(\sqrt {\bf{2}} \)are rational, which is a contradiction, therefore using this Poincare map we will never obtain the same point twice.