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Bifurcation Diagrams

  1. Sep 17, 2011 #1
    The problem statement, all variables and given/known data

    For each of the following equations sketch the bifurcation diagram, determine type of bifurcation, and find the critical value of r.

    ẋ = rx + cosh(x)

    ẋ = x(r - sinh(x))

    ẋ = rx - xe-x2

    The attempt at a solution

    Fixed points satisfy

    f'(x) = r + sinh(x) = 0 ⇒ x* = arcsinh(-r) = -arcsinh(r).

    f'(x) = r - sinh(x) - xcosh(x) = 0.

    f'(x) = r + 2xe-x2 - e-x2 = 0.
     
  2. jcsd
  3. Sep 18, 2011 #2
    Fixed points satisfy

    ẋ = rx + cosh(x) = 0

    ẋ = x(r - sinh(x)) = 0 ⇒ x* = 0, x* = arcsinh(r)

    ẋ = rx - xe-x2 = 0 ⇒ x* = 0, x* = ±√[-ln(r)].
     
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