I have multiple time series data that I have generated by varying the initial conditions infinitesimally. I now want to calculate the Lyapunov exponent to identify the sensitivity to initial conditions of the dynamic system that generated the data.
I am really confused on how to approach this as, the guides on [1]http://www.r-bloggers.com/logistic-growth-s-curves-bifurcations-and-lyapunov-exponents-in-r and [2]http://hypertextbook.com/chaos/43.shtml seem like they are using a single time series.. (as far as I understand)
The lya calculation on [2] is like:
for (i in 1:length(x)) {
lya[i] <- sum(log(abs(x[i]-(2*x[i]*XI[i,]))))/length(x)
}
which is quite confusing as it calculates the mean of just orbit for a single r i.e.: 0.5 then it appears to be calculating the mean of orbit corresponding to the logistic regression with r=0.5 over the first 500 iterations.. I don't see how this implementation actually compares different orbits...
If I have x orbits (from a dynamic system that I do not have an equation for, but just the data) than how should I go about calculating the Lyapunov exponent?
Thanks