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I want to plot eigenvalues of below time-delay system in MATLAB $$\dot x(t)= x(t)-a x(t-T)$$ T is delay

what I did until now is first I find the constitutive equation of this system as follow: $$ \lambda-1+ae^{-\lambda T}=0$$ then I write $$ \sigma +jw -1+ae^{- (\sigma +jw) T}$$ then I separate this into to equation $$ \sigma -1+ae^{- \sigma T } cos(wT)=0\\ w -ae^{- \sigma T } sin(wT)=0 $$ finally for finding them I try solve this two equation I write below code

  clc
clear all
close all
sigma=0;
a=2;
T=1/3;
j=1;
for i=1:1000
    sigma=-0.01+sigma;
    func=@(w)sigma-1+a*exp(-sigma*T)*cos(w*T);
    x0=0;
   w(i) = fzero(func,x0); 
   o(i)=  w(i)-a*exp(-sigma*T)*sin(w(i)*T);
                          if abs(o(i))<0.01
                nm(j) =w(i);
                j=1+j;
            end
         end
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  • $\begingroup$ there should be infinite number of solutions, corresponding with harmonics of 1/T. if you want, e^-jwT can be approximated (for a limited freq range) using Pade approximant, but that might not be what you are required to do here $\endgroup$
    – Pete W
    Nov 16, 2021 at 17:57

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