f1[t_, a_] := {Sin[3*t + a], Tan[t + a]}
f2[t_, a_] := {Sin[Exp[1.3*Sin[t - a]]] - Pi, Tan[t]}
f3[t_, a_] := {-Sin[Exp[1.3*Sin[t - a]]] + Pi, Tan[t]}
f4[t_, a_] := {Sin[2*t - a] + 1.75*Pi, Tan[t] + 1.2*Sin[t]}
f5[t_, a_] := {-Sin[2*t - a] - 1.75*Pi, Tan[t] + 1.2*Sin[t]}
g[a_] := ParametricPlot[{f1[t, a], f2[t, a], f3[t, a], f4[t, a],
   f5[t, a]}, {t, 0, 2 Pi},
  PlotRange -> {{-3 Pi, 3 Pi}, {-1.5 Pi, 1.5 Pi}}, Axes -> False,
  PlotStyle -> {{Blue, Thickness[0.01]}}]
duration := 600
For[i = 0, i <= duration, i++,
 Export["out" <> ToString[i] <> ".eps", g[E + 2 Pi*i/duration],
  ImageResolution -> 600]]