Considering the tent map where $x_{n+1} = f(x_{n})$ and $f(x)$ is defined as
$$ f(x)= \begin{cases} \mu x, & 0 \leq x\leq \frac{1}{2} \\ \mu - \mu x, & \frac{1}{2}\leq x\leq 1 \\ \end{cases} \\0 \leq \mu \leq 2 $$ I was wondering how I could go about classifying the bifurcation that occurs when $\mu = 1$.
I know that when $0\leq\mu<1$, there exists one stable fixed point ($x^*=0$). And when $1<\mu\leq2$, we retain the original $x^*=0$, but it is now unstable, and a new fixed point emerges at $ x^* = \frac{\mu}{\mu + 1}$, which is also unstable.
This behaviour doesn't seem to fit into any classification that I know of, and searching online did not produce any results. The exchange of stability for the trivial fixed point makes me lean towards a transcritical classification, but nothing of the sort happens for the second critical point, making me uncertain. The other idea was to classify it as a pitchfork bifurcation, since at $\mu=1$, a new fixed point emerges. The issue here is that theres no symmetry, so again I am back to the drawing board.
Any insight to how I can figure this out would be much appreciated.