# Different color representation for same circle

h[x0_] := Module[{b = x0}, f[z_] := 2 z^2;
z[t_] := Cos[t] + I Sin[t];
w = ComplexExpand[f[z[t]]];
p1 = ParametricPlot[{Re[z[t]], Im[z[t]]}, {t, 0, b},PlotRange -> All, PlotStyle->{Red}];
p2 = ParametricPlot[{Re[w], Im[w]}, {t, 0, b}, PlotRange -> All, PlotStyle -> {Green}];
Show[p1, p2]]
Manipulate[h[p], {p, \[Pi]/10, 2 \[Pi], \[Pi]/10}]


We know that $$f[z]:=2z^2$$ represents a circle when $$z$$ moves from $$0$$ to $$\pi$$. Now, when $$z$$ moves from $$\pi$$ to $$2\pi$$, the function $$f[z]$$ takes one round of circle and complete the circle.

When the first round is completed by $$f[z]$$, the line continues to be shown in green color. I want $$f[z]$$ to change color when it starts on a new round.

Similarly, when it starts its third (or fourth and so on) round its color should change every time.

What can I do?

You need to specify a color function that changes every $$2\pi$$:

z[t_] := Cos[t] + I Sin[t];

Manipulate[
ParametricPlot[
{Re[z[t]], Im[z[t]]},
{t, 0, tmax},
ColorFunction -> Function[{x, y, u}, ColorData[97, Quotient[u, 2 Pi]]],
ColorFunctionScaling -> False,
PlotStyle -> Thickness[0.1]
],
{tmax, 2 Pi, 10 Pi}
] 