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Discrete mapping and period doublings

  1. Mar 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi all.

    I am given the following discrete mapping: [itex]x_{n+1}=f(x_n)=x_n+r-x_n^2[/itex] for r>0.

    Objective: Find the r, where a period doubling takes occurs.

    Attempt: First I find the fixed points: These are [itex]x=-\sqrt{r}[/itex] (which is unstable for all r) and [itex]x=\sqrt{r}[/itex] (which is stable for 0<r<1).

    This is where I am stuck. I thought that I could find the r, where the first period-2 cycle occurs, but I do not think this is a period doubling. In that case I should find out when the period-4 cycle occurs, which seems like a long task (surely there must be an easier way).

    Can you shed some light on this?

    Thanks for helping.

    Best regards,
    Niles.
     
  2. jcsd
  3. Mar 20, 2009 #2
    Ok, I solved it. I have to find the point r, where there is a flip-bifurcation (i.e. where the gradient f'(x)=-1, where x is our stable fixpoint). This is the critical gradient, and there are (often, and in this case) period doublings when a flip bifurcation occurs.
     
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