For the first question, were asked what the first step of calculating the Lyapunov exponent from a scalar time-series data set is
And this is to embed the data
That is, to perform delay-coordinate embedding on the time series
Question 2 asks if Wolfs algorithm for calculating Lyapunov exponents uses the variational equations, and this is false
The variational equation is a way to get at the full spectrum of Lyapunov exponents if you have the governing equations, which you usually do not
Wolfs algorithm is a way of calculating the Lyapunov exponents given a trajectory of a dynamical system
For Question 3, were asked why its hard to calculate the Lyapunov exponent from data
And the answer is E, all of A through D above
You dont get to drop points where you want them in real-world data
The data points are almost always noisy
The data may not be long enough to fully cover the attractor
And the sampling rate of the data may not be high enough to capture all of the dynamics on the attractor
These are all certainly reasons that it would be hard to calculate the Lyapunov exponent, but certainly there are more reasons as well
Finally, for Question 4, consider this plot of the stretching factor produced by Kantzs algorithm
Part a asks if this curve has a visible scaling region, and it does
Region B would be considered a scaling region of this plot, so the answer to a is yes
Part b asks what the Lyapunov exponent of this trajectory is
The slope of a line fitted to the curve in region B
That is, the slope of a line fitted to the scaling region of this plot