Let us consider the discrete time dynamical system $x_{t+1} = f(x_{t}) = a - 5x_{t}^2$.
It is given that $a>-\frac{1}{20}$ and $x_{t} \in [-1,1]$.
I am trying to see at which value of parameter $a$, the map undergoes the period doubling bifurcation of period 1 point.
Say we calculate the value of $a$, for period one point then $f(x^*) = x^*$ implying $a-5x^2 =x$ implying $x^* = \frac{-1 \pm \sqrt{1 + 20a}}{10}$ Period doubling bifurcation occurs when the eigenvalues of the Jacobian matrix crosses $-1$. Hence, here its the derivative $f'(x) = -1$ at $x=x^*$ so, through this I obtain $a = \frac{3}{20}$. Is this correct?
Next, for period -2 period doubling bifurcation, we need $f^2(x) = x$, $f^2(x)$ is given by $a-5a^2 -125x^4 + 50ax^2$.
For the period 2 solution we need $f^2(x) = x$ implying $a-5a^2 -125x^4 + 50ax^2 = x$.
and for the period doubling bifurcation we need $(f^2(x))' = -1$ implying $-500x^3 + 100ax = -1$. I am struggling to find the value of $a$ where the period doubling bifurcation occur?