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Sensitivity to initial conditions

  1. May 19, 2015 #1
    Let M = {x1, x2, x3, ...} U {p} be a perfect metric space.
    Let f be continuous, taking M to M with f(xn) = xn+1 and f(p) = p.
    I would like to know if this dynamical system is necessarily sensitive to initial conditions.
     
  2. jcsd
  3. May 21, 2015 #2

    WWGD

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    What do you mean by sensitive to initial conditions? How do you define it?
     
  4. May 21, 2015 #3
    There exists d > 0, such that for each x in M and V a neighborhood of x there is y in V and a positive integer n with the property:
    the distance between f(n)(x) ( the nth iterate of f) and f(n)(y) is > d.
     
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