Question:
Consider the discrete map
$$x_{n+1} = \mu \sin (\pi x_n) \qquad \qquad x \in [0,1]$$
where $\mu \in [0,1]$ is a parameter.
Find the value of $\mu$ for which a bifurcation occurs at the point $x=0$
State the normal form at the bifurcation and identify the type of bifurcation.
Attempt:
Let $f(x) = \mu \sin (\pi x)$.
For the first part, I computed
$$|f'(0)| = 1 \implies |\mu \pi| = 1 \implies \mu = \frac 1\pi$$
so a bifurcation occurs at $\mu = 1/\pi$. Is this correct?
For the second part, I let $\mu^* = \mu - 1/\pi$ so that a bifurcation occurs at $\mu^* = 0$ for $x=0$. Moreover, Taylor expanding about $x_n=0$ we get
$$x_{n+1} = (\pi\mu^*- 1)x_n - \frac{\pi^2(\pi\mu^*- 1)}{6}x_n^3 + \mathcal O\big(x_n^5 \big)$$
Is this correct? I don't recognise this type of bifurcation though? Looks like a pitchfork but it isn't? (Pitchfork is supposed to be $\, x_{n+1} \sim rx_n - x_n^3 \,$ I think?)
Any hints would be much appreciated. Thanks!
You get an equivalent iteration using $y_n=\sqrt{\frac{1+πμ^∗}6}πx_n$ where $$ y_{n+1}=(1+πμ^∗)y_n-y_n^3+O(y_n^3) $$ which has the desired normal form.