Bouncing in a circle- chaos visualisation

g = 0.1; k = 0; TMax = 110; num = 2; vxs = {0.001,
  0.0012}; r = 10; tail = TMax;
f[x_, y_] := Sqrt@(x^2 + y^2);
toColour[{x_, y_}] :=
  Darker [Hue[ArcTan[x, y]/(2 Pi)], 0.9 (x^2 + y^2)/r^2];
path[x0_, y0_] := NDSolve[{x'[0] == 0, y'[0] == 0,
    x''[t] ==
     If[f[x[t], y[t]] > r, -f[x[t], y[t]], 0] x[t]/
       Sqrt[x[t]^2 + y[t]^2] - k x'[t],
    y''[t] == -g - k y'[t] +
      If[f[x[t], y[t]] > r, -f[x[t], y[t]], 0] y[t]/
       Sqrt[x[t]^2 + y[t]^2]

    , x[0] == x0, y[0] == y0}, {x[t], y[t]}, {t, 0, TMax}];

paths = Table[path[v], {v, vxs}];
pts = Partition[
   Flatten[Table[{x, y}, {x, -r, r, r/15}, {y, -r, r, r/15}]], 2];
pts = Select[pts, 0 < Norm[#] < r &];
cpaths = Table[path[p[[1]], p[[2]]], {p, pts}];
chaos[T_] := Module[{j}, Show[
   Graphics[{
     PointSize[Large],
     Table[
      {toColour[First[{x[t], y[t]} /. cpaths[[i]] /. {t -> T}]],
       Point[pts[[i]]]}
      , {i, Length[cpaths]}]
     }
    ], ImageSize -> 145
   ]
  ];
 Manipulate[chaos[\[Tau]], {\[Tau], 0, 110}]