# Lyapunov coefficient of Lorenz System

• I
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):  And here's the logarithm of the difference between the two (meaning the difference of the lengths of the vector in coordinate space): 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[f][t] == ecf, Derivative[p][t] == ecp,
Derivative[d][t] == ecd, f == 0.001, p == 0.,
d == 1} /. par, {f, p, d}, {t, t0, tf},
MaxSteps -> 10000000];
Plot[Evaluate[(f[t] /. solnum1), {t, t0, tf}], PlotRange -> All]

solnum2 =
NDSolve[{Derivative[f][t] == ecf, Derivative[p][t] == ecp,
Derivative[d][t] == ecd, f == 0.001 + eps,
p == 0. + eps, d == 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]