Leading polynomial approximation to a renormalization fixed point

Suppose $g(x)=1+c{x}^d+O(x^{d+2})$, with even $d$ and $c<0$. Expanding the period-doubling renormalization operator exactly at zero gives $a^{d-1}{g}^{\prime }(1)=1$. Approximating globally by $1+c{x}^d$ gives $c\approx a-1$ and $g^{\prime }(1)\approx dc$, hence $d(a-1)a^{d-1}\approx 1$. For $d=2$, $a\approx (1-\sqrt{3})/2$, $g(x)\approx 1-(1+\sqrt{3})x^2/2$ and $\alpha \approx 1+\sqrt{3}$. The coefficient relation is a polynomial-truncation approximation, not an exact identity for the full analytic fixed point.

// Target: numerical-analysis.bigb