w0 = 0.05;
w = 0.05;
\[CapitalDelta]g = 0.5;
U = 0.01;
For[g = 0.001, g < 0.15, g = g + 0.001,
solution =
Solve[8*\[CapitalDelta]g^2*y*x^3 +
x*(2*w + 4*w0*\[CapitalDelta]g^2 - U*y) + 2*g*Sqrt[1 - y^2] ==
0, {x}];
x = x /. solution;
b = Sqrt[
1 - y^2]*(w0/2 - 2*w0*\[CapitalDelta]g^2*x^2 +
2*w*\[CapitalDelta]g^2*y + U/2*x^2 -
2*U*\[CapitalDelta]g^2*x^4 + U*\[CapitalDelta]g^2*(1 - y^2)/2 -
2*U*\[CapitalDelta]g^2*y^2) + 2*g*x*y;
a = RootSearch[b == 0, {y, -1, 1}];
Print[g, a]]