x = Array[0 &, {10000}];
y = Array[0 &, {10000}];
[Sigma] = Array[0 &, {10000}];
S = 0
x[[1]] = 0.1;
y[[1]] = x[[1]];
[Alpha] = 0.4*Pi;
Do[x[[i + 1]] = x[[i]]*Cos[[Alpha]] - Sin[[Alpha]]*(y[[i]] - x[[i]]^2);
y[[i + 1]] =
x[[i]]*Sin[[Alpha]] + Cos[[Alpha]]*(y[[i]] - x[[i]]^2);[Sigma][[i]] =
ArcCos[y[[i + 1]]/(Sqrt[x[[i + 1]]^2 + y[[i + 1]]^2])] -
ArcCos[y[[i]]/(Sqrt[x[[i]]^2 + y[[i]]^2])], {i, 9999}]
ListPlot[Table[{x[[i]], y[[i]]}, {i, 1, 10000}]]
ListPlot[Table[[Sigma][[i]], {i, 1, 9999}]]
Do[S = (S + [Sigma][[i]])/10^4, {i, 9999}]