 Solution

Apply the Poincaré-Lindstedt method to the truncated system from part (b). Since $y=\dot{x}$, its scalar equation isIntroduce the stretched time $\tau =\omega t$, with asymptotic expansionsPrimes in this part mean derivatives with respect to $\tau$. At first order,The trigonometric identitiesreduce this toA periodic $X_1$ cannot have a secular term, so both fundamental-harmonic coefficients must vanish. Taking $A>0$ givesIn particular $\mu >0$ is necessary. Once these conditions hold, a bounded particular correction isadding fundamental sine and cosine terms adjusts the phase and the first correction to the amplitude. Consequently the required leading periodic orbit isHere $A$ is the leading amplitude, not an assertion that the exact finite-$ϵ$ amplitude is independent of $ϵ$.

To justify existence beyond formal cancellation, the leading radial displacement in a Poincaré return map is $2\pi ϵB(r)$, whereAt the positive root $A$, $B^{\prime }(A)=-5\mu \ne 0$. The implicit function theorem, applied after dividing the displacement by $ϵ$, continues this root to a periodic orbit for sufficiently small positive $ϵ$. Its radial Floquet multiplier is $1-10\pi \mu ϵ+O({ϵ}^2)$, 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.