A1 = 2;
A2 = 1;
ω = 3;
δ = π/4;
T = 2 π/ω;

x[t_] := A1 Cos[ω t]

y[t_,phase_] := A2 Cos[ω t + phase]

Animate[
ParametricPlot[{x[t],y[t,δ]},{t,0,T},
Frame -> True, FrameLabel -> {x,y},
RotateLabel -> False, PlotRange -> {{-A1,A1},{-A2,A2}},
Epilog -> {PointSize[Medium],Point[{x[t],y[t,δ]}]}],
{t,0,T}]

Manipulate[
Animate[
ParametricPlot[{x[t],y[t,phase]},{t,0,T},
Frame -> True, FrameLabel -> {x,y},
RotateLabel -> False, PlotRange -> {{-A1,A1},{-A2,A2}},
Epilog -> {PointSize[Medium],Point[{x[t],y[t,phase]}]}],
{t,0,T}],
{phase,0,2 π}
]

AA1 = 2;
AA2 = 1;
ω1 = 3;
ω2 = 2 ω1;
δδ = π/4;
TT = 2 π/ω1;

xx[t_] := AA1 Cos[ω1 t]

yy[t_] := AA2 Cos[ω2 t + δδ]

Animate[
ParametricPlot[{xx[t],yy[t]},{t,0,TT},
Frame -> True, FrameLabel -> {x,y},
RotateLabel -> False, PlotRange -> {{-A1,A1},{-A2,A2}},
Epilog -> {PointSize[Medium],Point[{xx[t],yy[t]}]}],
{t,0,TT}]