Discrete mapping and period doublings

[Niles]

Homework Statement

Hi all.

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

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

Attempt: First I find the fixed points: These are $x=-\sqrt{r}$ (which is unstable for all r) and $x=\sqrt{r}$ (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.


Niles.

 

[Niles]

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.