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\title{A Planar Discrete Dynamical System}

\begin{document}
\maketitle

Let $s,\eta$ be two parameters, for example $s =1, \eta = 0.1$.

Let $ \CF'_1 : (x,y) \to ( x',y')$ be a function from $[0,1] \times [0,1]$ to $[0,1] \times [0,1]$, where
\begin{align}
    &x' = \frac{x}{1+s(1-x)(1-y)},\\
    &y' = \frac{y}{1-s(1-x)(1-y)}.
\end{align}

Given this pair of $\left( (x,y), ( x',y') \right)$ as defined above, define the function $$\CF_1 : (x,y) \to (x'',y'')$$ as following:

\begin{align}
    &x'' = \frac{x}{1+\eta(1-x)(1-y')},\\
    &y' =  \frac{y}{1-\eta(1-x')(1-y)}.
\end{align}

Note that since $( x',y')$ is determined by $(x,y)$ through $\CF'_1$, thus $\CF_1$ is still a map from $[0,1] \times [0,1]$ to $[0,1] \times [0,1]$.
\\
\\
Let $\CF'_2 : (\tilde{x},\tilde{y}) \to ( \tilde{x}',\tilde{y}')$ be a function from $[0,1] \times [0,1]$ to $[0,1] \times [0,1]$, where
\begin{align}
    &\tilde{x}' = \frac{\tilde{x} (1+s\tilde{y}) }{1+s \tilde{x}\tilde{y}},\\
    \\
    &\tilde{y}' = \frac{\tilde{y} (1+s\tilde{x}) }{1+s \tilde{x}\tilde{y}}.
\end{align}
\\
\\
Given this pair of $\left( (\tilde{x},\tilde{y}), ( \tilde{x}',\tilde{y}') \right)$ as defined above, define the function $$\CF_2 : (\tilde{x},\tilde{y}) \to (\tilde{x}'',\tilde{y}'')$$ as following:

\begin{align}
    &\tilde{x}'' = \frac{\tilde{x}(1+\eta \tilde{y}' )}{1+\eta \tilde{x}\tilde{y}' },\\
    \\
    &\tilde{y}'' = \frac{\tilde{y}(1+\eta \tilde{x}' )}{1+\eta \tilde{x}'\tilde{y} }.
\end{align}
\\
\\
The dynamical system is defined as $\CF_2 \circ \CF_1 : [0,1] \times [0,1] \to [0,1] \times [0,1]$. \textcolor{red}{We want to prove $\CF_2 \circ \CF_1$ will converge to some fixed point.} 

When $s =1, \eta = 0.1$, an orbit of $\CF_2 \circ \CF_1$ looks like following:

\begin{figure}[h]
    \centering
    \includegraphics[width=0.8\textwidth]{1.jpg}
    \caption{An orbit of $\CF_2 \circ \CF_1$}
    \label{KL-OMWU}
\end{figure}



\end{document}