= Solution
Adjoin $\mu_{n+1}=\mu_n$. At the origin the stable direction is the $x$ axis, with <Floquet multiplier> zero, while the $y$ and parameter directions have <Floquet multiplier> one. By the <centre manifold theorem for a discrete dynamical system>, write the <extended centre manifold> as
$$
x=h(y,\mu),\qquad h(0,0)=0,\qquad Dh(0,0)=0.
$$
On it the scalar map is $Y=(1+\mu)y-\mu h(y,\mu)+y^3$. Its discrete <centre-manifold invariance equation> is
$$
h(Y,\mu)=\mu y.
$$
Put $h=h_2+O(\rho^3)$ with $\rho=|y|+|\mu|$. Since $Y=y+O(\rho^2)$, the quadratic equation is simply $h_2(y,\mu)=\mu y$. Thus
$$
\boxed{x=\mu y+O(\rho^3).}
$$
In particular, setting the stable variable identically to zero would lose the leading parameter correction. The reduced map through total degree three is
$$
\boxed{y_{n+1}=y_n+\mu y_n-\mu^2y_n+y_n^3+O(\rho^4).}
$$
In the natural <pitchfork bifurcation> scaling $\mu=O(y^2)$ this has leading form $y_{n+1}=y_n+\mu y_n+y_n^3$. Its nonzero <fixed points> satisfy $y^2\sim-\mu$, and their reduced <fixed-point multipliers> satisfy $1-2\mu+O(\mu^2)>1$ for small $\mu<0$. The transverse <Floquet multiplier> remains inside the <unit circle>. Therefore \b[the <bifurcation> at $\mu=0$ is a <subcritical pitchfork bifurcation>: the origin attracts for small negative $\mu$, each nonzero branch is a <saddle fixed point of a map>, and the origin is a <saddle fixed point of a map> for small positive $\mu$].
An exact check uses the full <Jacobian matrix> on either nonzero branch. Its <trace> is $T_*=1-2\mu+3\mu^2$ and its <determinant> is $D_*=\mu^2$. Thus
$$
p_*(1)=1-T_*+D_*=2\mu(1-\mu)<0
$$
whenever a nonzero branch exists. Also $p_*(0)=\mu^2>0$, so one real <Floquet multiplier> lies in $(0,1)$ and the other in $(1,\infty)$. This proves that each nonzero branch is a <saddle fixed point of a map> throughout $\mu<0$ and $\mu>1$, in particular near the requested <bifurcation>.
Back to article page
