= Solution
The <Jacobian matrix> at the origin and its <characteristic polynomial> are
$$
M_0=\begin{pmatrix}0&\mu\\-\mu&1+\mu\end{pmatrix},\qquad
p(\lambda)=\lambda^2-(1+\mu)\lambda+\mu^2.
$$
The <fixed-point multipliers> are
$$
\lambda_\pm=\frac{1+\mu\pm\sqrt{(1-\mu)(1+3\mu)}}{2}.
$$
By the <Jury stability criterion>, both lie strictly inside the <unit circle> exactly when
$$
|\mu^2|<1,\qquad p(1)=\mu(\mu-1)>0,\qquad p(-1)=\mu^2+\mu+2>0.
$$
The last condition always holds, since $\mu^2+\mu+2=(\mu+1/2)^2+7/4$. Therefore \b[the origin is <asymptotically stable> exactly for $-1<\mu<0$ among the hyperbolic parameter values].
For completeness, its <linear stability> throughout the parameter line is as follows. When $\mu<-1$, the complex-conjugate <Floquet multipliers> have <modulus> $|\mu|>1$, so the origin repels. When $-1<\mu<-1/3$, their <modulus> is $|\mu|<1$, so it attracts with oscillation. At $\mu=-1/3$ the repeated <Floquet multiplier> is $1/3$, still attracting. For $-1/3<\mu<0$, both <Floquet multipliers> are real and in $(0,1)$. For $0<\mu<1$, $p(1)<0$ and $p(0)>0$, so one <Floquet multiplier> lies in $(0,1)$ and the other exceeds one: the origin is a <saddle fixed point of a map>. For $\mu>1$, the complex-conjugate <Floquet multipliers> have <modulus> $\mu>1$, so it repels.
Possible changes at the <unit circle> are exhausted by $p(1)=0$, $p(-1)=0$, or a nonreal pair with product one. They give
$$
\boxed{\mu=-1,\quad0,\quad1.}
$$
At $\mu=0$ the <Floquet multipliers> are $0,1$; the nonzero <fixed point> branches in part (a) meet the origin, and part (c) identifies a <subcritical pitchfork bifurcation>. Indeed at $\mu=0$ the exact restricted map is $y\mapsto y+y^3$, which makes the origin unstable despite its neutral linear direction.
At $\mu=-1$ the <Floquet multipliers> are $i,-i$: this is a 1:4 <strong resonance of a planar map>, coinciding with the change from repelling to attracting as $\mu$ increases. A generic nonresonant <Neimark–Sacker bifurcation> conclusion would be unjustified. At $\mu=1$ there is a double <Floquet multiplier> $+1$; the matrix has a nontrivial <Jordan block>, and the two further nonzero <fixed point> branches meet the origin. <Linear stability> alone is inconclusive at these unit-modulus boundary values. The repeated <Floquet multiplier> at $\mu=-1/3$ is strictly inside the <unit circle> and gives no <bifurcation>.
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