= Solution
A <steady-state bifurcation> is a change in the local equilibrium branches or their stability as a parameter varies. At such a bifurcation the <Jacobian matrix> at the equilibrium has a zero <eigenvalue>, so the implicit function argument for a unique persistent local equilibrium fails.
For this system,
$$
J=\begin{pmatrix}1-y^2-3x^2&-2xy\\-2xy&\mu-2y-x^2\end{pmatrix}.
$$
At $(0,\mu)$ its eigenvalues are $1-\mu^2,-\mu$; at $(1,0)$ they are $-2,\mu-1$. Both have one simple zero eigenvalue at $\mu=1$, with the other eigenvalue negative. Interior equilibrium branches, obtained below, meet these boundary equilibria there, establishing the actual bifurcations rather than merely a necessary eigenvalue test.
$$
\boxed{\mu=1:\quad\sigma(J_{(0,1)})=\{0,-1\},\quad\sigma(J_{(1,0)})=\{-2,0\}.}
$$
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