I'm working through a problem with Lorenz attractors. I've successfully plotted one solution with parameter $\sigma_1 = 28$ and another with parameter $\sigma_2=28.0001$ showing the chaotic behavior when a parameter is perturbed slightly.
I now need to find the logarithm of the vector norm (length) of the difference between the two solutions. What does the vector norm mean in this context? How do I find it?
My attempt: I tried considering both solutions as vectors themselves and simply subtracting them, but I don't think this is correct.
You probably want to observe the phenomenon of exponential divergence or quantify it via Lyapunov exponents. In that case what you want is the vector norm of the difference between the states of the systems at a given time (not the entire solutions), usually the Euclidean distance.
Suppose your solutions are $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$, then the Euclidean norm $d$ of the difference at a given time $t$ is:
$$d(t) = \sqrt{ (x_1(t)-x_2(t))^2 + (y_1(t)-y_2(t))^2 + (z_1(t)-z_2(t))^2 }$$
This distance exhibits an exponential growth with $t$ for sufficiently close initial conditions (or parameters as in your case) and a sufficiently small $t$.