3 アニメーション

gifs={};
Do[
If[t==0, 
dotsout={PointSize[0.02]};
dotsin={PointSize[0.01]}
];

xin[t_]:={2Sin[A t+phi]+2.5,-t/Pi-0.5};
yin[t_]:={-t/Pi-0.5,2Sin[B t]+2.5};
lissajues[t_]:={2Sin[A t+phi]+2.5,2Sin[B t]+2.5};
baseliney:=Line[{{-5,2.5},{5,2.5}}];
baselinex:=Line[{{2.5,-5},{2.5,5}}];

AppendTo[dotsout,Point[lissajues[t]]];
AppendTo[dotsin,{Point[xin[t]],Point[yin[t]]}];
gifs=AppendTo[gifs,
Graphics[
{dotsin,
{Green,dotsout},
{Red,Line[{xin[t],lissajues[t]}]},{Red,Line[{yin[t],lissajues[t]}]},
baselinex,
baseliney},PlotRange->{{-5,5},{-5,5}}
]
],
{A,1,2,1},{B,2,3,1},{phi,0,Pi/2,Pi/4},{t,0,2Pi,Pi/40}]
ListAnimate[gifs]

Directory[]
Export["lissajues2.gif",gifs]