= Solution
The fixed points are $(0,0)$ and $(\pm|\mu|,\mu^2)$. At the origin the <eigenvalues> are $\mu^2$ and $-1$, so it is a saddle. At either nonzero point the characteristic <polynomial> is
$$
\lambda^2+\lambda+2\mu^2.
$$
They are stable nodes for $0<|\mu|<1/(2\sqrt2)$, degenerate nodes at equality, and stable spirals above it.
The stable manifold of the origin is exactly $x=0$. Write the <stable manifold> in the unstable direction as $y=ax^2+bx^4+O(x^6)$. Graph invariance gives
$$
a=\frac1{1+2\mu^2},\qquad
b=\frac{2a^2}{1+4\mu^2}.
$$
Thus $W^u$ has
$$
\boxed{y=\frac{x^2}{1+2\mu^2}
+\frac{2x^4}{(1+2\mu^2)^2(1+4\mu^2)}+O(x^6).}
$$
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