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I am not so familiar with Mathematica programming. Here is my problem. I have defined the parameters in the first step:

$G = 0.6, \Gamma = 0.5, r_1 = 1, r_2 = 1, \theta_1 = \pi/4, \theta_2 = \pi/4$, and $t=0$

Then, I have defined some functions:

Δ = G - Γ; 
ω = Sqrt[-4 + G^2 + 2 G Γ + Γ^2];

r11 = (E^(t Δ) (ω Cosh[t ω] + (G + Γ) Sinh[t ω]))/ω;
r12 = (2   E^(t Δ) Sinh[t ω])/ω;
s11 = ((G + Γ + ω) E^( (Δ + ω) (t - u )))/(2 ω) + ((-G - Γ + ω) E^( (Δ - ω) (t - u )))/(2 ω);
s12 = (E^( (Δ + ω) (t - u )))/ ω - (E^( (Δ - ω) (t - u )))/ ω;

Ia = Cosh[t Δ]^2 Cosh[t ω]^2 Sinh[r1]^2 + 2 Cosh[t Δ] Cosh[t ω]^2 Sinh[r1]^2 Sinh[t Δ] + Cosh[t ω]^2 Sinh[r1]^2 Sinh[t Δ]^2 + (2 G Cosh[t Δ]^2 Cosh[t ω] Sinh[r1]^2 Sinh[t ω])/ω + (2 Γ Cosh[t Δ]^2 Cosh[t ω] Sinh[r1]^2 Sinh[t ω])/ω // Chop;

var = r11^2 Cosh[r1]^2 + r12^2 Cosh[r2]^2 - 
    2 r11^2 Cos[θ1] Cosh[r1] Sinh[r1] + r11^2 Sinh[r1]^2 + 
    r12^2 Sinh[r2]^2 +
    r12^2 Cos[θ2] Sinh[2 r2] + 
    Integrate[2 G s11^2 + 2 s12^2 Γ // Expand, {u, 0, t}] // Chop;

At this step, I should change the definition of G as $G = 0.6/(1 + I_a/2)$ and the new time is $t = t + 0.001$. Now we should turn back to the beginning of the codes and repeat this loop until $t=3$. Finally, I should plot $Ia$ and $-Log[var]$ from $t= 0$ to $3$. If $G$ remains constant, it is easy to plot, but I need to calculate the new $G$ at each step and replace it. I will appreciate it if you help me.

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Use a Table to collect your points and then extract and plot.

G=0.6; Γ=0.5; r1=1; r2=1; θ1=π/4; θ2=π/4;
points = Table[
  Δ=G-Γ; ω=Sqrt[-4+G^2+2 G Γ+Γ^2]; 
  r11=(E^(t Δ)(ω Cosh[t ω]+(G+Γ)Sinh[t ω]))/ω; 
  r12=(2 E^(t Δ)Sinh[t ω])/ω; 
  s11=((G+Γ+ω)E^((Δ+ω)(t-u)))/(2 ω)+((-G-Γ+ω)E^((Δ-ω)(t-u)))/(2 ω); 
  s12=(E^((Δ+ω)(t-u)))/ω-(E^((Δ-ω)(t-u)))/ω; 
  Ia=Cosh[t Δ^2] Cosh[t ω]^2 Sinh[r1]^2+2 Cosh[t Δ]Cosh[t ω]^2 Sinh[r1]^2 Sinh[t Δ]+
    Cosh[t ω]^2 Sinh[r1]^2 Sinh[t Δ]^2+(2 G Cosh[t Δ]^2 Cosh[t ω]*
    Sinh[r1]^2 Sinh[t ω])/ω+(2 Γ Cosh[t Δ]^2 Cosh[t ω] Sinh[r1]^2*Sinh[t ω])/ω//Chop; 
  var=r11^2 Cosh[r1]^2+r12^2 Cosh[r2]^2-2 r11^2 Cos[θ1] Cosh[r1] Sinh[r1]+
    r11^2 Sinh[r1]^2+r12^2 Sinh[r2]^2+r12^2 Cos[θ2] Sinh[2 r2]+ 
    Integrate[2 G s11^2+2 s12^2 Γ // Expand, {u, 0, t}] //Chop;
  G=0.6/(1+Ia/2);
  {{t, Ia}, {t, -Log[var]}}, {t,0,3,.001}];
Iapoints = Map[First, points];
varpoints = Map[Last, points];
ListPlot[Iapoints]
ListPlot[varpoints]

enter image description here

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Please check all this very carefully to make sure there are no errors

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