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