Lyapunov coefficient of Lorenz System

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  • Thread starter carllacan
  • Start date
  • #1
274
3
Hi.

I'm trying to get the Lyapunov coeficient for a Lorenz System (namely, a laser, using the Lorenz-Haken model) but I'm not getting the plots that would be expected. This is how two trajectories with near identical initial conditions behave (only one of the variables):

lyapunov.png

lyapunov.png


And here's the logarithm of the difference between the two (meaning the difference of the lengths of the vector in coordinate space):
lyapunov3.png


It looks like it grows exponentially at first, but then it stops. Why is this?

Here is my code (Mathematica):
Code:
t0 = 0;
tf = 250;

eps = 10^-5;

ecf = s (p[t] - f[t]);
ecp = -p[t] + d[t] f[t];
ecd = b (r - d[t] - f[t] p[t]);

par = {s -> 3., b -> 1, r -> 30};

solnum1 =
  NDSolve[{Derivative[1][f][t] == ecf, Derivative[1][p][t] == ecp,
     Derivative[1][d][t] == ecd, f[0] == 0.001, p[0] == 0.,
     d[0] == 1} /. par, {f, p, d}, {t, t0, tf},
   MaxSteps -> 10000000];
Plot[Evaluate[(f[t] /. solnum1), {t, t0, tf}], PlotRange -> All]

solnum2 =
  NDSolve[{Derivative[1][f][t] == ecf, Derivative[1][p][t] == ecp,
     Derivative[1][d][t] == ecd, f[0] == 0.001 + eps,
     p[0] == 0. + eps, d[0] == 1 + eps} /. par, {f, p, d}, {t, t0,
    tf}, MaxSteps -> 10000000];
Plot[Evaluate[(f[t] /. solnum2), {t, t0, tf}], PlotRange -> All]

Plot[Log[Sqrt[((f[t] /. solnum1) - (f[t] /. solnum2))^2 + ((p[t] /. solnum1) - (p[t] /. solnum2))^2 + ((d[t] /. solnum1) - (d[t] /. solnum2))^2]], {t, t0, tf},
PlotRange -> All]
 

Answers and Replies

  • #2
18,668
8,628
Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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