Suppose a system of 2 equations defined as:
$$\begin{cases} x_{n+1}&=f_x(x_n,y_n) \\ y_{n+1}&=f_y(x_n,y_n) \end{cases}$$
where initial conditions for $x_0$ and $y_0$ are defined, and $x_n,y_n\in\Bbb R$, $f_x,f_y:\Bbb R^2\to \Bbb R$.
I found that if $f_x(x,y)=\cos(x+y)+\sin(x-y)$ and $f_y(x,y)=\cos(x-y)+\sin(x+y)$. Then the system becomes
$$\begin{cases} x_{n+1}&=\cos(x_n+y_n)+\sin(x_n-y_n)\\ y_{n+1}&=\cos(x_n-y_n)+\sin(x_n+y_n) \end{cases}$$
which for any starting values for $x_0$ and $y_0$ it will produce the following points(plotted $x$ horizontally and until $x_{1.000.000}$):
which creates a rather interesting structure(I think its called an attractor), and I'd love to hear more about these systems, but couldn't find anything. Any help would be appreciated. Thanks.
EDIT:
If the system is
$$\begin{cases} x_{n+1}&= \cos(x_n^2+y_n^2) \\ y_{n+1}&= \sin(x_n^2-y_n^2) \end{cases}$$
The points would be the following:
Are these coming from some differential equations?
Yes, such recurrences have been studied extensively and are still subject to resear in the field of Dynamical Systems, discrete time dynamical systems in particular. In this field the behaviour of repeated applications of a function (in your case $\mathbb R^2 \to \mathbb R^2$) to some inputs is studied.