= Stability at a nondegenerate flip bifurcation
{title2=$c_3+c_2^2>0$}
For a $C^4$ real map $f(u)=-u+c_2u^2+c_3u^3+O(u^4)$, the second iterate is
$$
f^2(u)=u-2(c_3+c_2^2)u^3+O(u^4).
$$
If $c_3+c_2^2>0$, 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 $f'(0)=-1$. 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>.
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