cycles described by $O_{1}$, the discrete dynamical system:$$\left\{
\begin{array}
[c]{c}%
x_{n+1}=x_{n}+2P(x_{n})+P(y_{n})+P(x_{n}-y_{n})\\
y_{n+1}=y_{n}+P(x_{n})+2P({y}_{n})-P(x_{n}-y_{n}).
\end{array}
\right.$$ If we write $$\left\{
\begin{array}
[c]{c}%
\varepsilon\varphi\left(  x,y\right)  =2P(x)+P(y)+P(x-y)\\
\varepsilon\gamma\left(  x,y\right)  =P(x)+2P({y})+P(y-x),
\end{array}
\right.$$ then we have $$\varphi\left(  x,y\right)  =\gamma\left(  y,x\right)  \text{,}%$$ and the iteration is a perturbation of the identity as $$\left[
\begin{array}
[c]{c}%
x_{n+1}\\
y_{n+1}%
\end{array}
\right]  =\left[
\begin{array}
[c]{cc}%
1 & 0\\
0 & 1
\end{array}
\right]  \left[
\begin{array}
[c]{c}%
x_{n}\\
y_{n}%
\end{array}
\right]  +\varepsilon\left[
\begin{array}
[c]{c}%
\varphi(x_{n},y_{n})\\
\varphi(y_{n},x_{n})
\end{array}
\right]  ,$$ that we can also write as $$X_{n+1}=F(X_{n})=X_{n}+\varepsilon\Omega(X_{n}), \label{Model}%$$ where $$X_{n+1}=\left[
\begin{array}
[c]{c}%
x_{n+1}\\
y_{n+1}%
\end{array}
\right]  ,$$ $$%
\begin{array}
[c]{c}%
F(X_{n})
\end{array}
=\left[
\begin{array}
[c]{cc}%
1 & 0\\
0 & 1
\end{array}
\right]  \left[
\begin{array}
[c]{c}%
x_{n}\\
y_{n},
\end{array}
\right]$$ and $$%
\begin{array}
[c]{c}%
\Omega(X_{n})
\end{array}
=\left[
\begin{array}
[c]{c}%
\varphi(x_{n},y_{n})\\
\varphi(y_{n},x_{n})
\end{array}
\right]  .$$

We now consider $P\left(  x\right)  =\varepsilon\sin x$, where $\varepsilon
=\frac{\alpha\mu}{8h^{2}}$ from hypothesis \[Perturb1\], explicitly,$$\begin{aligned}
\varphi\left(  x,y\right)   &  =2\sin x+\sin y+\sin\left(  x-y\right) \\
\gamma\left(  x,y\right)   &  =\sin x+2\sin y+\sin\left(  y-x\right)  .\\
&\end{aligned}$$

Analysis of the model
=====================

Fixed points and local stability
--------------------------------

In this section, we analyze the model (\[Model\]) obtained in the previous section. In a nutshell, in this section, we see that the system is 