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Lyapunov exponent -- Numerical calculations

  1. Jun 30, 2015 #1
    In computational physics is very often to calculate largest Lyapunov exponent. If largest Lyapunov exponent ##LE## is positive there is chaos in the system, if it is negative or zero there is no chaos in the system. But what can we say about some certain value of ##LE##. For example ##LE_1=-0.2## and ##LE_2=-0.4##. What is the difference between those particular values? Could we say something in small scales? Thanks for the answer.
  2. jcsd
  3. Jul 3, 2015 #2
    The more negative the LE is, the more quickly the trajectory returns to its unperturbed path in that dimension of phase space.
  4. Jul 3, 2015 #3
    Thank you for the answer. Is it necessary that if LE is negative motion is periodic? And when LE is zero motion is quasiperiodic?
  5. Jul 3, 2015 #4
    Too many variables in play for a simple answer. Conservative or not? Winding numbers? Dimensions?

    A negative LE does not imply a periodic orbit.
  6. Jul 3, 2015 #5
    Ok thanks. But periodic orbits imply that LE is negative. Right?
  7. Jul 3, 2015 #6
    Not always. There are cases of unstable periodic orbits where the LE is positive.

    Try a google search for unstable periodic orbits. There are a lot of examples.
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