Natural Equation

frame[a_] := Module[{soln},
   \[Phi][s_] :=
    Sum[a (-1)^i Sin[i s + a]/(i (i + 1)), {i, 1, 100, 2}];
   soln = NDSolve[{
      x[0] == 0, y[0] == 0,
      y'[s] == Sin[\[Phi][s]],
      x'[s] == Cos[\[Phi][s]]
      },
     {x[s], y[s]},
     {s, 0, 2 Pi}
     ];
   ParametricPlot[{x[s], y[s]} /. soln /. {s -> st}, {st, 0, 2 Pi},
    ImageSize -> {300, 300}, Axes -> None,
    PlotStyle -> Directive[Thick, RGBColor[0.2, 0.1, 0.3]]]
   ];
Manipulate[frame[a], {a, 0, 5 Pi}]