$\lambda_2 = \frac{dF(f(x))}{dx}\vert_{x=x_1} = F’(F(x_1))F’(x_1)=F’(x_2)F’(x_1)$
From [2022-01-25#Stability of the 2 cycle|previous class], stability is attained for $|4+2r-r^2| < 1$, that is, $3<r<1+\sqrt{6}\approx3.449$
At birth of $P_4$, it is locally the same as $P_2$
$\frac{r_m - r_{m-1}}{r_{m+1} - r_m}\to 4.669 \equiv \delta$ As $m\to \infty$
Accumulation point of an infinite number of period doublings.
Not just logistic map
Typical of period doubling route to chaos
Seen in experiments in variety of physical systems
Hence, 1D map applies to higher dimensional systems.