 Stability at a nondegenerate flip bifurcation

For a $C^4$ real map $f(u)=-u+c_2{u}^2+c_3{u}^3+O(u^4)$, the second iterate isIf $c_3+c_2^2>0$, sufficiently small nonzero $u$ retains its sign under $f^2$ and strictly decreases in absolute value. Its iterates therefore converge to zero, proving local asymptotic stability even though $f^{\prime }(0)=-1$. If $c_3+c_2^2<0$, the second iterate moves small positive points away from zero, proving instability. The zero-coefficient case requires higher-order terms. With strictly stable transverse directions, the centre manifold theorem for a discrete dynamical system applies this scalar test to a higher-dimensional discrete dynamical system.