1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lorentz Chaos - The 'Butterfly Effect'

  1. Apr 28, 2015 #1
    1. The problem statement, all variables and given/known data

    Given the lorentz system for ##\sigma=10, b = \frac{8}{3}, r = 28##, and ##x(t)## from the first lorentz system, show that we can solve for y(t) and z(t) for the modified lorentz system by finding ##\dot E##.


    2013_B1_Q3.png

    2. Relevant equations


    3. The attempt at a solution

    I have found the 3 fixed points. They are at the origin ##(0,0,0)##, and ##C^{+} = \left( \sqrt{b(r-1)}, \sqrt{b(r-1)}, r-1 \right)## and ##C^{-} = \left(-\sqrt{b(r-1)}, -\sqrt{b(r-1)} , r-1 \right)##. For ##r = 28##, all three points are unstable.

    It turns out that the points ##C^{+}, C^{-}## are only stable for ##1 < r < 25##.

    For a dynamical system, the Lyapunov exponent ##\lambda## is related to the trajectory in phase space by
    [tex]|\delta V(t) | = |\delta V_0| e^{\lambda t} [/tex]
    So does this mean that for ##\lambda > 0## these trajectories are replled from one unstable point to another unstable point? I think this is the 'butterfly effect' described somewhere.

    Also, I have re-expressed the equations:
    [tex]\dot e_x + \dot x = \sigma \left[ (e_y - e_x) + (y-x) \right] [/tex]
    [tex]\dot e_y + \dot y = rx - (e_y + y) - x(e_z + z) [/tex]
    [tex]\dot e_z + \dot z = x(e_y + y) - b(e_z + z) [/tex]
    [tex]\dot E = \frac{2}{\sigma} e_x \dot e_x + 2 e_y \dot e_y + 2 e_z \dot e_z[/tex]

    I'm not sure what the question wants..
     
    Last edited: Apr 28, 2015
  2. jcsd
  3. May 3, 2015 #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?
     
  4. May 3, 2015 #3
    My problem with this question stems from the confusion of what they want. I tried re-reading this question but it still didn't make sense to me. Would appreciate some insight from others.
     
  5. May 4, 2015 #4
    any insight on last part?
     
  6. May 7, 2015 #5
    any help on the chaos bit?
     
  7. May 10, 2015 #6
    bumpp on chaos
     
  8. May 14, 2015 #7
    butterfly bumpping
     
  9. May 16, 2015 #8
    bump on lorentz chaos
     
  10. May 17, 2015 #9
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Lorentz Chaos - The 'Butterfly Effect'
  1. Lorentz transformation (Replies: 1)

  2. Lorentz Transformations (Replies: 29)

Loading...