Evolution

Clear[ycoords];
ycoords[0] = Accumulate[RandomReal[NormalDistribution[0, 0.2], 20]];
ycoords[n_] :=
  ycoords[n] =
   ycoords[n - 1] + RandomReal[NormalDistribution[0, 0.05], 20];
Clear[xcoords];
xcoords[0] = Accumulate[RandomReal[NormalDistribution[1, 0.1], 20]];
xcoords[n_] :=
  xcoords[n] =
   Sort[xcoords[n - 1] + RandomReal[NormalDistribution[0, 0.1], 20]];
k = 100; ListPlot[
 Table[RotationMatrix[2 Pi i/(k + 2)].(# + {10, 0}) & /@
   Partition[
    Riffle[MovingAverage[xcoords[i], 2],
     MovingAverage[ycoords[i], 2]], 2], {i, 0, k}], Axes -> None,
 Joined -> True, InterpolationOrder -> 3,
 PlotStyle -> Table[ColorData["RedBlueTones"][i/k], {i, 0, k}],
 ImageSize -> {500}, AspectRatio -> 1, Background -> Black]