= Solution
Apply the <Poincaré-Lindstedt method> to the truncated system from part (b). Since $y=\dot x$, its scalar equation is
$$
\ddot x+x=\epsilon\left(5\mu\dot x-3x^2\dot x+2x\dot x^2-2\dot x^3\right).
$$
Introduce the stretched time $\tau=\omega t$, with <asymptotic expansions>
$$
\omega=1+\epsilon\omega_1+O(\epsilon^2),\qquad
x(t)=X(\tau)=A\cos\tau+\epsilon X_1(\tau)+O(\epsilon^2).
$$
Primes in this part mean <derivatives> with respect to $\tau$. At first order,
$$
X_1''+X_1=2\omega_1A\cos\tau-5\mu A\sin\tau
+3A^3\cos^2\tau\sin\tau+2A^3\cos\tau\sin^2\tau+2A^3\sin^3\tau.
$$
The <trigonometric identities>
$$
\cos^2\tau\sin\tau=\frac{\sin\tau+\sin3\tau}{4},\qquad
\cos\tau\sin^2\tau=\frac{\cos\tau-\cos3\tau}{4},\qquad
\sin^3\tau=\frac{3\sin\tau-\sin3\tau}{4}
$$
reduce this to
$$
X_1''+X_1=
\left(2\omega_1A+\frac{A^3}{2}\right)\cos\tau
+\left(-5\mu A+\frac{9A^3}{4}\right)\sin\tau
-\frac{A^3}{2}\cos3\tau+\frac{A^3}{4}\sin3\tau.
$$
A periodic $X_1$ cannot have a <secular term>, so both fundamental-harmonic <coefficients> must vanish. Taking $A>0$ gives
$$
\boxed{A^2=\frac{20\mu}{9},\qquad \omega_1=-\frac{A^2}{4}=-\frac{5\mu}{9}.}
$$
In particular $\mu>0$ is necessary. Once these conditions hold, a bounded particular correction is
$$
X_1=\frac{A^3}{16}\cos3\tau-\frac{A^3}{32}\sin3\tau;
$$
adding fundamental sine and cosine terms adjusts the <phase> and the first correction to the <amplitude>. Consequently the required leading <periodic orbit> is
$$
\boxed{x=A\cos(\omega t)+O(\epsilon),\quad
y=-A\omega\sin(\omega t)+O(\epsilon),\quad
\omega=1-\frac{5\mu\epsilon}{9}+O(\epsilon^2).}
$$
Here $A$ is the leading <amplitude>, not an assertion that the exact finite-$\epsilon$ <amplitude> is independent of $\epsilon$.
To justify existence beyond formal cancellation, the leading radial displacement in a <Poincaré return map> is $2\pi\epsilon B(r)$, where
$$
B(r)=\frac{5\mu}{2}r-\frac98r^3.
$$
At the positive root $A$, $B'(A)=-5\mu\ne0$. The <implicit function theorem>, applied after dividing the displacement by $\epsilon$, continues this root to a <periodic orbit> for sufficiently small positive $\epsilon$. Its radial <Floquet multiplier> is $1-10\pi\mu\epsilon+O(\epsilon^2)$, so it is attracting. These statements apply to the truncated system with fixed $\mu>0$; its interpretation as a local approximation to the original three-dimensional system also requires $\mu$ small, so that $A=O(\sqrt\mu)$ stays near the <Hopf bifurcation>.
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