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Lorenz Equations and Chaos

  1. Apr 29, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that there is a certain ellipsoidal region E of the form rx2 + σy2 + σ(z-2r)2 ≤ C such that all trajectories of the Lorenz equations eventually enter E and stay in there forever.

    2. Relevant equations

    Lorenz Equations:

    [itex]\dot{x} = \sigma (y - x)[/itex]

    [itex]\dot{y} = rx - xz - y[/itex]

    [itex]\dot{z} = xy - bz[/itex]


    V is the volume.

    3. The attempt at a solution

    [itex] V = rx^2 + \sigma y^2 + \sigma(z-2r)^2 [/itex]

    [itex]\dot{V} = 2rx\dot{x} + 2\sigma y \dot{y} + 2 \sigma(z - 2r) \dot{z}[/itex]

    [itex]\dfrac{1}{2} \dot{V} = rx(\sigma y - \sigma x) + \sigma y (rx - xz - y) + (\sigma z - 2 \sigma r)(xy - bz)[/itex]

    [itex]\Rightarrow \dfrac{1}{2 \sigma} \dot{V} = -rx^2 - y^2 -bz^2 + 2bz[/itex]


    [itex]\Rightarrow \dot{V} < 0[/itex]

    Therefore, all trajectories of the Lorenz equations eventually enter E and stay in there forever.



    I am not sure I approached this problem correctly, and the additional challenge problem states to try and obtain the smallest possible value of C with this attracting property. Did I miss something in my workthrough above involving C? If I subtract it from both sides of the original inequality and involve it in the workthrough involving the change in volume, isn't the value of C irrelevant given that its derivative is zero (being that it's a constant) ?
     
  2. jcsd
  3. Apr 29, 2013 #2
  4. Apr 29, 2013 #3
    That seems to help quite a bit. Now I'm only going half as crazy as I was. Thanks a lot!
     
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