= Solution
For parameter-dependent <centre manifold theorem>, append $\dot\mu=0$. At $(x,y,\mu)=(0,0,0)$ the <eigenvalues> are $0,-1,0$: the $y$ direction is strongly stable and $(x,\mu)$ span the extended centre directions. An <extended centre manifold for a parameter> is an invariant <graph of a function> $y=h(x,\mu)$ tangent to $y=0$; nearby trajectories approach its reduced dynamics along stable fibers. The stable <eigenvalue> remains separated from zero, so local bifurcation and stability are governed by this reduction. This is the parameter-extension content of the requested stable/centre-manifold discussion.
The <centre-manifold invariance equation> is
$$
h_x(\mu x+xh-x^3)=-h+h^2-x^2.
$$
Reflection symmetry permits an invariant <graph of a function> with $h(-x,\mu)=h(x,\mu)$ and $h(0,\mu)=0$. At quadratic order the left side vanishes and the right side is $-h-x^2$, giving $h=-x^2+O(|\mu|x^2+x^4)$. More explicitly its first mixed coefficient is $2\mu x^2$, found by equating the $\mu x^2$ terms. The reduced equation is
$$
\boxed{\dot x=\mu x-2x^3+O(|\mu|x^3+x^5).}
$$
Thus the origin is locally asymptotically stable for $\mu<0$, becomes a saddle for $\mu>0$, and creates two stable equilibria $x=\pm\sqrt{\mu/2}+O(\mu^{3/2})$, $y=-\mu/2+O(\mu^2)$. This is a \b[supercritical <pitchfork bifurcation>]. The exact nearby nonzero branches provide a check: the equilibrium equations give
$$
y=1-\sqrt{1+\mu},\qquad x^2=1+\mu-\sqrt{1+\mu},
$$
which are real and nonzero for small positive $\mu$. The omitted branch $y=1+\sqrt{1+\mu}$ is not near the origin. At $\mu=0$ the leading $-2x^3$ also attracts, although at an algebraic rather than exponential rate.
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