= Solution
The map <normal forms> at multiplier $+1$ are given in the numbered subparts below. For the multiplier $-1$ problem, the second iterate instead exposes the emerging period-two cycle.
For the multiplier $-1$ problem, write $x=X+\alpha/2$. To the required weighted order, with $X=O(\mu^{1/2})$, the shifted map is $X_{n+1}=-X_n+bX_n+\gamma X_n^2+\delta X_n^3$, where $b=\beta+\alpha\gamma$. Composing it with itself cancels the quadratic term. The cubic contribution is $-2\delta X^3-2\gamma^2X^3$, while the linear correction is $-2bX$. Consequently the <quadratic-cubic flip criticality> calculation gives
$$
\boxed{\widehat\mu=-2(\beta+\alpha\gamma),\qquad A=2(\delta+\gamma^2).}
$$
Thus the second iterate has normal form $X_{n+2}=X_n+\widehat\mu X_n-AX_n^3$, up to terms of weighted order at least four. This is the asymptotic interpretation of the stated remainder under $X=O(\mu^{1/2})$; a uniformly valid local expansion about the exact <fixed point> also retains mixed parameter remainders.
Assuming $\widehat\mu$ crosses zero transversely and $A\ne0$, the small nonzero fixed points of the second iterate satisfy $X^2\sim\widehat\mu/A$. They form a <period-two orbit>. For $A>0$ they exist on the side $\widehat\mu>0$, where the original fixed point has become unstable, and the second-iterate multiplier is $1-2\widehat\mu+o(\widehat\mu)$, so the cycle is stable. The <period-doubling bifurcation> is therefore \b[supercritical precisely when $\delta+\gamma^2>0$]. Equality is a degenerate case requiring higher-order terms.
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