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Bifurcations in maps

  1. May 16, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove carefully that the following map has a horseshoe for one positive value of [itex]\mu[/itex]
    xn+1 = xn2 - [itex]\mu[/itex]
    For what positive values of [itex]\mu[/itex] will the map have a horseshoe?
    2. Relevant equations

    3. The attempt at a solution
    xn+1 = xn2 - [itex]\mu[/itex] = g(xn,[itex]\mu[/itex])
    ∂g/∂x = 2x
    → turning pt at 2x = 0 → turning pt at x=0

    Intersects the x axis at 0 = x2 - [itex]\mu[/itex]
    x = ±√[itex]\mu[/itex]

    I am not even sure if I am on the right track at all. It feels very wrong so any help would be greatly appreciated
  2. jcsd
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