I'm trying to understand how symmetry transformations give us indication of what kind of bifurcation occurs in a particular system, and I'm currently following Ghrist et. al.'s monograph on Knots & Links in 3-D flows, which is a little sketchy at places.
In particular, for pitchfork bifurcation he writes:
Consider $f:\mathbb{R}^1\rightarrow \mathbb{R}^1$ which is generic in the class of maps which is invariant under the symmetry transformation $x\mapsto -x$ e.g. $x\mapsto x + \mu x - x^3$. Then, by symmetry, the origin must be a fixed point for all $\mu$. In this case there is a pitchfork bifurcation at $\mu=0$.
For the period-doubling bifurcation, he writes:
Let $f_{\mu}:\mathbb{R}^1\rightarrow \mathbb{R}^1$ be a generic map whose derivative satisfies $f'_0(0)=-1$, e.g. $x\mapsto - x - \mu x + x^3$. Then, the bifurcation at $\mu=0$ is called a period-doubling bifurcation.
My questions: 1) I learnt about pitchfork bifurcation in the context of ODEs, where there were a bunch of other conditions necessary for a pitchfork bifurcation to take place, yet they aren't mentioned here. It's unclear to me why a generic real-valued map that is invariant under the symmetry transformation results in a pitchfork bifurcation when we vary the parameters. For instance, $g(x)=x$ is invariant under the symmetry transformation since $-x \mapsto -x = g(-x)$
2) I don't really know what he means by symmetry transformation in this case. Does he mean that the function is odd? But in this case, isn't the period-doubling bifurcation also invariant under the symmetry transformation. For if, $f(x) = - x - \mu x + x^3$, then $f(-x)= - (x -\mu x + x^3) = -f(x)$, no? Wouldn't that make it a pitchfork bifurcation under Ghrist's definition above? Yet that can't be right... I'm basically confused about how period-doubling transformations are symmetry breaking...
I know these are probably really basic questions, but any help would be much appreciated!