Suppose I have a 2-dimensional map $ x_{n+1} = f(x_{n},y_{n}), \quad y_{n+1} = g(x_{n},y_{n})$ , and a number of initial conditions $ (x_{0}, y_{0})$ in some given range. I take each initial condition and iterate under the map a certain number of times. How do I generate a list of all the results and a phase portrait that shows all the iterates of all the initial condition?

For example. Take $ x_{n+1} = x_{n} + 2y_{n}, \quad y_{n+1} = y_{n}$ with $ x_{0} \in [1,3]$ and $ y_{0} \in [2,5]$ , with these intervals partitioned with a step size of $ 0.2$ , i.e. I have $ 150$ initial conditions.

I assume I should use Recurrence Table and ListPlot, but beyond that I’m stuck.