4
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If I have 2000 data points for the following equation ,How to find out the Lyapunov exponent for different values of [Tau] .This is a delay differential equation i.e infinite dimensional system.

b = 1;
V0 = 300;
I0 = 0.001;
N1 = 7/4;
p1 = (\[Pi]*b*N1)/V0;
a1 = 1.001; a2 = 0.123; a3 = -3.622*10^-3; b1 = 0.001959; b2 = 0.031; 
b3 = 0.003241; G = 0.5*10^-5; \[CapitalOmega] = 0; C1 = p1/2000;
f = 1*10^3;
Is = 0.0;
\[Tau] = 15;
NL = 1000;

sol1[t_] = 
 NDSolve[{x'[t] - (I0*p1)/(
      2*C1)*((a1*x[t - \[Tau]] - G/(p1*I0)*x[t]) - 
        3/4*x[t - \[Tau]]^3*a2 - 5/8*a3*x[t - \[Tau]]^5) - 
     Is*p1*Cos[y[t]] == 0, 
   y'[t] - \[CapitalOmega] - (I0*p1)/(
      2*C1*x[t])*(b1*x[t - \[Tau]] + 3/4*x[t - \[Tau]]^3*b2 - 
        5/8*b3*x[t - \[Tau]]^5) + (Is*p1)/(2*C1*x[t])*Sin[y[t]] == 0, 
   x[0] == 0.003, y[0] == 0.001}, {x, y}, {t, 0, NL}]
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  • $\begingroup$ Not implemented in MMA except for this, which is pre-MMA8 and I didn't make it work. Try using other implementations, e.g. TISEAN, .exe programmes or Wolf et. al fortran implementation. $\endgroup$ – corey979 Sep 9 '16 at 9:55
  • $\begingroup$ So there is no way to find out Lyapunov Exponent of DDE using Mathematica? $\endgroup$ – Udichi Sep 9 '16 at 10:06
  • $\begingroup$ There could be different approaches depending on whether you want a method to work on a time series (see @corey979's response and this yet-unanswered question) or a method to work inside the NDSolve (see Marco Sandri's package, but it's not clear it it will work on a delay equation without modification). Judging from the multiple unanswered questions on this topic, looks like something a lot of people want. $\endgroup$ – Chris K Sep 9 '16 at 11:29
  • $\begingroup$ In another post ( mathematica.stackexchange.com/questions/125613/… ) I have given a program where I have tried to find out LE using NDSolve but the problem of that code is for different values of [Tau] I am getting the same result which is not correct. Can I get help to modify that code so that I can get different LE for different [Tau] . $\endgroup$ – Udichi Sep 9 '16 at 16:51
  • $\begingroup$ What definition of Lyapunov Exponent are you using for the delay differential equation? $\endgroup$ – bbgodfrey Sep 13 '16 at 18:56

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