Consider the two-dimensional map $$ \begin{aligned} X_{n+1} &= X_n + Y_n + r\left(X_n - X^3_n\right)\\ Y_{n+1} &= Y_n + r\left(X_n - X^3_n\right) \end{aligned} $$ where $r$ is a control parameter. Find the fixed points of this map as a function of the control parameter $0 < r < 4$.
How do you find the fixed points of this map? I have no idea here how to proceed when there is a cubic term.
For a $2$D map $\begin{cases}x_{n+1}=f(x_n,y_n)\\y_{n+1}=g(x_n,y_n)\end{cases}$ the fixed point (or, critical point) are given by solution of the system of equation $\begin{cases}f(x,y)=x\\g(x,y)=y\end{cases}.$