Walker Attractor

map[{u_, v_}] := {1, 1} - v {Sin[Pi u/2], Cos[Pi u/2]};
pts = RandomReal[{0, 1}, {1000, 2}];
n = 2000;
squares = Table[
   pts = N@Join[
      s Table[{0, t/n} , {t, 0, n}],
      s Table[{ t/n, 1}, {t, 0, n}],
      s Table[{1, 1 - t/n}, {t, 0, n}],
      s Table[{1 - t/n, 0}, {t, 0, n}]
      ];
   Map[# + (1 - s) {1, 1}/2 &, pts],
   {s, 0.1, 1, 0.1}];
f[pts_] := Map[map, pts];
ptsLists = Table[NestList[f, pts, 20], {pts, squares}];
frame[i_] :=
  With[{\[Iota] = If[i < 5,
      If[i == 0, 0,
       1 + Sum[0.5 + 0.5 Tanh[5 (i - j - 0.5)], {j, 5}]],
      5 + (i - 5)^1.5]},
   Graphics[{
     EdgeForm[],
     Table[
      Module[{ps},
       ps =
        ptsLists[[j, Floor@\[Iota]]] (1 - Mod[\[Iota], 1]) +
         ptsLists[[j, Floor@\[Iota] + 1]] Mod[\[Iota], 1];
       ps =
        Map[(# - {0.5, 0.5}) (0.7 + 0.3 (1 - Cos[2 Pi \[Iota]])/2) &,
         ps];
       {
        FaceForm[ColorData["RedBlueTones"][j/Length@squares]],
        Polygon[ps.RotationMatrix[Pi \[Iota]/2]]
        }
       ]
      , {j, Length@squares, 1, -1}]
     },
    PlotRange -> 0.5
    ]
   ];
frames = Table[frame[i], {i, 1, 11.1, 11.1/87}];
Export["walker.gif", frames]