I want to calculate the Lyapunov indices for a system - problem is that I can’t get DynamicalSystems.jl to work (described here).
Therefore, I decided to do it a much more crude way by writing out the steps myself. Now, if I understand it right, the algorithm will compare the systems (with initial distance d_0) at a certain time. We note the distance in the 2 trajectories here as d_1, move the perturbed system back to d_0 distance from the evolved unperturbed trajectory, run the evolution for the next interval, repeat.
In order to do this, I decided to solve my original system for some tspan. Now, getting d_1 is easy enough - solve the same system with different initial conditions and subtract the 2 timeseries. So, for d_1, lets say I evolve the system for time, t_j. I am not sure what to do to get d_2, but I thought of the following way - solve the same system again but this time I use,
u_0 = [ x = sol[1, j] + d_0 ]
tspan = (t_j, t_k)
So, I choose the initial value of my variable this time as the value of the unperturbed variable at t_j plus the deviation, d_0. Now, I evolve this system from t_j to some other time t_k. Repeat the process.
I have not yet tried this (I will as soon as I get the chance), but I would like to know what the people here, who are familiar with Lyapunov exponents and so on, think of the validity and the viability of the process.
Thanks in advance!!