= Solution
Adjoin $\dot\mu=0$ and put $w=z+\mu$. The <extended centre manifold> is a graph $w=H(x,y,\mu)$ tangent to $w=0$ at the origin. The equations become
$$
\dot x=y,\qquad \dot y=-x+5\epsilon\mu y-5\epsilon y w,\qquad \dot w=-w+x^2,\qquad \dot\mu=0.
$$
Here $\epsilon>0$ is fixed. The <centre-manifold invariance equation>, obtained by the <chain rule>, is
$$
H_x y+H_y(-x+5\epsilon\mu y-5\epsilon yH)=-H+x^2.
$$
Write $H=H_2+O(\rho^3)$, where $\rho=|x|+|y|+|\mu|$ and $H_2$ is a homogeneous quadratic <polynomial>. To quadratic order the <centre-manifold invariance equation> is
$$
y\partial_xH_2-x\partial_yH_2=-H_2+x^2.
$$
The terms $\mu x,\mu y,\mu^2$ have zero <coefficients>: their equations are homogeneous, and the operator $1+y\partial_x-x\partial_y$ is invertible on these quadratic <polynomials>. For $H_2=ax^2+bxy+cy^2$, comparison of <coefficients> gives
$$
-b=1-a,\qquad 2a-2c=-b,\qquad b=-c.
$$
Hence $a=3/5$, $b=-2/5$ and $c=2/5$, giving
$$
\boxed{z=-\mu+\frac35x^2-\frac25xy+\frac25y^2+O(\rho^3).}
$$
Substitution into the $y$ equation gives
$$
\boxed{\dot x=y,\qquad
\dot y=-x+\epsilon(5\mu y-3x^2y+2xy^2-2y^3)+O(\rho^4).}
$$
The constant in the <big O notation> may depend on fixed $\epsilon$. Thus the equations printed in part (b) are the retained terms of the local <Taylor series>, rather than an exact identity for the full reduced <vector field>.
For the <Hopf bifurcation> classification in part (a), set $\mu=0$ and use $x=r\cos\theta$, $y=-r\sin\theta$. The radial derivative is $\dot r=(x\dot x+y\dot y)/r$. Its cubic angular mean is
$$
\frac{\epsilon}{r}\left\langle-3x^2y^2+2xy^3-2y^4\right\rangle
=-\epsilon r^3\left(\frac38+\frac68\right)=-\frac{9\epsilon}{8}r^3.
$$
Here $\langle\cdot\rangle$ denotes the mean over one revolution. Removing the oscillatory cubic terms by a <near-identity transformation> gives this same cubic radial coefficient in the <Hopf normal form>. The linear radial coefficient is $5\epsilon\mu/2$.
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