Manipulate[ParametricPlot[{a1 Cos[omega t + phi], a2 Cos[omega2 t+ phi2]},{t,0,10 Pi}],{a1,1,4},{a2,1,4},{omega,1,4},{omega2,1,4}, {{phi,Pi/2},-Pi,Pi},{{phi2,Pi/2},-Pi,Pi}]

Animate[ParametricPlot[{Cos[ 2t ], Cos[ t]},{t,t0,t0+.05}, PlotRange -> {-1,1}, PlotStyle -> Thick], {t0,0.001,10Pi},AnimationRunning->False, AnimationRate-> .05]