OBM 2004 Problem 6

Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$ . If $P=(x;y)\in R^2$ , define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$ .

The set $per(a;b)$ of the periodic points of $f_{a,b}$ is the set of points $P\in R^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$ .

Fix $b$ . Prove that the set $A_b=\{a\in R \mid per(a;b)\neq \emptyset\}$ admits a minimum. Find this minimum.