Lorenz system directions of fastest growth

[Fek]

Homework Statement

A Lorenz system is given by
x' = sy - sx
y' = 3sx - y - xz
z' = xy - bz

In the vicinity of point x = y = 2s, z=4s , for s>>b, show motion is aligned with the y,z plane (sub these values in and x' is 0). By considering the evolution of vectors [2,0,1+sqrt(5)] , [0,1,0], [1+sqrt(5),0,-2] , or otherwise, estimate the directions in which perturbations grow and contract fastest.

Homework Equations

I've calculated Jacobian + Jacobian transpose / 2 is:
-s 2s s
2s -1 0
s 0 b

The Attempt at a Solution

No idea, other than get rid of b then operate with the matrix on the perturbation vectors. The vectors are clearly not eigenvectors though. Setting b = 0 does not return any sensible eigenvectors or values, and trying to solve for the eigenvalues with b small, but non-zero descends into a mess.

Many thanks for any help

 

[mark726]

you can offer!
Thank you for your post. The Lorenz system is a well-studied system in dynamical systems and chaos theory. In the vicinity of the given point, the x' equation becomes 0, which means that the motion is aligned with the y,z plane. This can also be seen by setting x=2s in the y' and z' equations, which results in a system of two equations in two unknowns (y and z).

To estimate the directions in which perturbations grow and contract fastest, we can use the Jacobian matrix as you have calculated. The eigenvalues of this matrix will give us information about the stability of the system. However, as you have noted, the perturbation vectors are not eigenvectors of this matrix. In this case, we can use the concept of principal directions to estimate the directions of fastest growth and contraction.

The principal directions are the directions in which the Jacobian matrix has the largest and smallest eigenvalues. To find these directions, we can use the method of diagonalization. First, we find the eigenvalues of the Jacobian matrix. For s>>b, we can neglect the b term and the eigenvalues are approximately -s and 2s. Next, we find the corresponding eigenvectors, which are [1,2,0] and [0,0,1], respectively. These are the principal directions in which perturbations will grow and contract fastest.

I hope this helps. Let me know if you have any further questions.