In my problem I need to plot roots with respect to a parameter. But the condition for root is det(Function)=0. This function cannot be solved symbolically. So I need to put in a parameter value and guess the value of the root. I am not able to get my desired result. Any help will be highly appreciated. Below is my Mathematica code.
Δ = 0*0.5;
λ = .5;
u = .3;
δ = (0.5*α)/ 1 - u);
a1 = λ^2 + Δ^2 + δ^2 - Ω^2
a2 = 2*Δ*λ
a3 = 2*Δ*δ
a4 = 2*δ*λ
L = a1 - a2*Sin[2*x] + a3*Cos[x] - a4*Sin[x]
a11 = -((Δ*Sin[x] - λ*Cos[x])^2/L)
a12 = -2 ((Δ*Sin[x] - λ*Cos[x]) (λ*
Sin[x] - Δ*Cos[x] - δ))/L
a13 = -2 Ω (Δ*Sin[x] - λ*Cos[x])/L
a21 = a12
a22 = -2 (λ*Sin[x] - Δ*Cos[x] - δ)^2/L
a23 = -I*2*Ω (λ*Sin[x] - Δ*
Cos[x] - δ)/L
a31 = -a13
a32 = -a23
a33 = -2 (1 + Ω^2/L)
ListPlot[Table[{Ω, Max[Table[If[(Im[Det[({
{(1 + u/(4 π)*NIntegrate[a11, {x, 0, 2 π}]),
u/(4 π)*NIntegrate[a12, {x, 0, 2 π}],
u/(4 π)*NIntegrate[a13, {x, 0, 2 π}]},
{
u/(4 π)*
NIntegrate[a12, {x, 0, 2 π}], (1 +
u/(4 π)*NIntegrate[a22, {x, 0, 2 π}]),
u/(4 π)*NIntegrate[a23, {x, 0, 2 π}]},
{-(u/(4 π))*
NIntegrate[a13, {x, 0, 2 π}], -(u/(4 π))*
NIntegrate[a23, {x, 0, 2 π}], (1 +
u/(4 π)*NIntegrate[a33, {x, 0, 2 π}])}
})]]
+ Re[Det[({
{(1 + u/(4 π)*NIntegrate[a11, {x, 0, 2 π}]),
u/(4 π)*NIntegrate[a12, {x, 0, 2 π}],
u/(4 π)*NIntegrate[a13, {x, 0, 2 π}]},
{
u/(4 π)*
NIntegrate[a12, {x, 0, 2 π}], (1 +
u/(4 π)*NIntegrate[a22, {x, 0, 2 π}]),
u/(4 π)*NIntegrate[a23, {x, 0, 2 π}]},
{-(u/(4 π))*
NIntegrate[a13, {x, 0, 2 π}], -(u/(4 π))*
NIntegrate[a23, {x, 0, 2 π}], (1 +
u/(4 π)*NIntegrate[a33, {x, 0, 2 π}])}
})]]) <= .0000000, λ, 0]
, {λ, 0, 8, 0.1}]]}, {Ω, 0, 8, 0.1}],
Joined -> True]
