Matlab代写|3.3 Lyapunov exponents for the Lorenz model


(5 points, 1 point for each of the subtasks (b)-(f))

Please work on “3.1 Introduction to the Lorenz system” first because the results of
that exercise are helpful for doing this exercise.

It is also a good idea to do the exercise “3.2 Lyapunov exponents for a toy model”
before this one and test the code that you write here on the toy model.

The three-dimensional Lorenz flow is given by

(a) Write a code that numerically computes the Lyapunov exponents for the Lorenz
attractor. You may find the following hints helpful:

∙ Use the “cooking recipe” given in the lecture notes (Section 10.5) to calculate
Lyapunov exponents. Following this method, you iterate the discretised equation
for the deformation matrix. At each time step you evaluate the Jacobian along your
solution trajectory (x(t), y(t), z(t)). The Lyapunov exponents are obtained from
the diagonal elements of the R-matrix using the QR-decomposition method.

∙ You may want to use the built-in function “QRDecomposition[]” in Mathematica.
Read the documentation of that function carefully. In particular, note that (at least in
Mathematica Version 10 and below) it returns for a given matrix M the pair
{Q , R}.

∙ For the parameter values studied by Lorenz and discussed in the exercise
“Introduction to the Lorenz model”, σ = 10, b = 8/3, r = 28, the Lyapunov
exponents are approximately λ ≈ 0.91, λ ≈ 0, λ ≈ −14.57.

∙ In the exercise “Introduction to the Lorenz model” you have derived an
expression for the sum of the Lyapunov exponents, λ + λ + λ , for general
parameter values. This expression is numerically more stable than the results for
the individual Lyapunov exponents and it is very useful for checking that your code
works properly.