You can export Animate directly. If you want to keep controls:
Clear[r];
r[t_]:=5+Sin[10 t];
curves=ParametricPlot[r[t] {Cos[t],Sin[t]},{t,0,2 Pi}];
anim=Animate[Show[curves,
Graphics[{PointSize[0.03],Point[r[t] {Cos[t],Sin[t]}]}]],{t,0,2 Pi}];
Export["anim.gif",anim,AnimationRepetitions->Infinity]
