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Tent map

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data
    The "tent-map" is given by: xn+1 = g(xn) where g(x) = 2x if 0 <= x<= 1/2 and g(x) = 2-2x if 1/2 < x <= 1
    a) Find the fixed points and their stability. Draw a cobweb plot of the tent map to demonstrate that your stability calculations are correct.
    b) Find a period-2 orbit, and compute its stability.
    c) It can be shown that the tent-map has period-n orbits for all n [itex]\in[/itex] N. Without doing any calculations explain why all of these periodic orbits must be unstable.


    2. Relevant equations



    3. The attempt at a solution
    Done a). Got x = 0 and x = 2/3, both unstable and this is shown in the cobweb plot.
    For b) I do xn+2 = xn and got 4 period-2 points, x = 0, x = 2/3, x = 4/7, x = 2/7, but I don't think these are right and how do you compute stability for these.
    For c) Not too sure what they are asking here, it's not som obvious to me.
    Any help on b) and c) would be great.
     
  2. jcsd
  3. Aug 25, 2011 #2

    micromass

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    Your period 2 points are correct.
    Now, how did you define stability for such a points?
     
  4. Aug 25, 2011 #3
    Is it just the same in as in part a)?
     
  5. Aug 25, 2011 #4
    I now get x = 0, 2/3, 2/5 and 4/5 as my period-2 orbits.
     
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