Question 1 asks us to write a program that performs delay-coordinate embedding on a time series
Here is a Matlab implementation of this code
Notice its very simple: its just two nested for loops that walk over the time series and place the elements in a matrix in the correct order
If you run this code using a tau of 8 and an m of 7 on the amplitude.dat time series, and we plot the zeroth element versus the second element, we get the following plot
This object does appear to be a chaotic attractor, so the answer to part a is Chaotic
Part b states that the dynamics used to generate the trajectory from the last problem was three-dimensional and had a capacity dimension of 2.1
What requirements does the Takens theorem place on m for a successful embedding of this time series?
This would be that m needs to be greater than 6
The Takens theorem says that the m should be chosen greater than 2 times the dimension of the original system, which in this case was three
So m > 6 is the answer
Notice that m = 7 would work, because 7 is greater than 6 here
However, the restrictions on the Takens theorem would state m greater than 6, not m equal to 7