# How to plot a phase portrait of a map for many initial points?

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 points $(x_{0}, y_{0})$ in some given range. I take each initial point and iterate 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 $0.2$ and, therefore, I have $150$ initial points.

I assume I should use RecurrenceTable and ListPlot, but beyond that I'm stuck.

NestList is built for this kind of thing. Define f and g:

f[x_, y_] := x + 2 y
g[x_, y_] := y


Then you can generate all your initial points with

Flatten[Table[{i, j}, {i, 1, 3, 0.2}, {j, 2, 5, 0.2}], 1]

(* {{-0.5, -0.5}, {-0.5, -0.4}, {-0.5, -0.3}, ...,
{0.5, 0.3}, {0.5, 0.4}, {0.5, 0.5}} *)


NestList can get you the iterates for an initical point, and you can Map (/ @) that over the list of inital points to get all the iterates:

points = NestList[{f[#[], #[]], g[#[], #[]]} &, #, 100] & /@
Flatten[Table[{i, j}, {i, 1, 3, 0.2}, {j, 2, 5, 0.2}], 1];

ListPlot[points] Notice that the iterated from each initial point are coloured differently by ListPlot.

Using the Duffing map for another example:

f[x_, y_] := y
g[x_, y_] := -0.2 x + 2.75 y - y^3
points = NestList[{f[#[], #[]], g[#[], #[]]} &, #, 1000] & /@
Flatten[Table[{i, j}, {i, -0.5, 0.5, 0.1}, {j, -0.5, 0.5, 0.1}], 1];
ListPlot[Flatten[points, 1]] If you want to get rid of transients -- which is often the case -- you can simply Drop the first n points to obtain a cleaner plot:

points = Drop[
NestList[{f[#[], #[]], g[#[], #[]]} &, #, 1000],
100] & /@
Flatten[Table[{i, j}, {i, -0.5, 0.5, 0.1}, {j, -0.5, 0.5, 0.1}], 1];
ListPlot[Flatten[points, 1]] 