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Lorenz system directions of fastest growth

  1. Jun 5, 2016 #1

    Fek

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    1. The problem statement, all variables and given/known data
    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.

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


    3. 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
     
  2. jcsd
  3. Jun 10, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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