iterated2Dmap.nb
To run this, put the cursor anywhere in the next cell.  Then type Shift-Enter, or hit the Enter key on the bottom right of the keyboard.  You can play with different values of a and b.
To see help for the commands, like Do, or anything that is capitalized, select the work and hit F1 (or go to the Help menu).

a = 0.49;
b = 0.95;
title = "Whitley map, (x, y) ↦ (y, a y + b - (x^2))\n a = "<>ToString[a]<>", b = "<>ToString[b];
fofxy[x_, y_] := {y, a y + b - x^2}; (* the Whitley map. *);

a = 1.4;
b = 0.3;
title = "Henon map, (x, y) ↦ (a - (x^2) + b y, x)\n a = "<>ToString[a]<>", b = "<>ToString[b];
fofxy[x_, y_] := {a - x^2 + b y,  x}; (* the Henon map. *);

(* Put the map you want to iterate last on the list *)

ic = {0,0};
nTransients = 1000;
nIterates = 10000;  (*Don't make this much larger than 50 000. It gets slow. *)
(* Only make changes above this line, unless you know what you're doing *)
f[v_] := fofxy[v[[1]], v[[2]] ];  (* maps points (lists) to points *)
pt = ic;
Do[pt = N@f[pt]
, {nTransients}
];
orbit = {pt};
Do[pt = f[pt];
AppendTo[orbit, pt]
, {nIterates}
];
figure = ListPlot[orbit, AspectRatio -> Automatic, Frame -> True, Axes -> False, PlotStyle -> PointSize[Tiny], PlotLabel ->title]

You can zoom in using this next cell.

Show[figure, PlotRange ->{{0.6,0.9}, {0.6,0.9}}]

If you like the figure, you can save it with this next cell.  Put in a unique name to replace “birdFeet.gif”, or the old figure will be over-written.  Customize your SetDirectory[  ] command: Delete my directory and go to the Insert menu and choose “File Path...”.

SetDirectory["C:\\Users\\jws8\\Dropbox\\classes\\MAT 667 2017.1\\Whitley Map"]
Export["birdFeet.gif", figure, ImageSize -> 300]