= Solution
For the perturbed amplitude equations, direct differentiation of the <first integral> gives
$$
F'=-\varepsilon u(\lambda_1+v^2)(\lambda_2+u^2+v^2).
$$
The axis connection contributes zero since $u=0$. On the unperturbed interior <heteroclinic orbit>, $u=\sqrt{3(a^2-v^2)}$ and $v'=2(a^2-v^2)$, so the first-order change is
$$
\Delta F=-\varepsilon\sqrt3\int_{-a}^a(\lambda_1+v^2)\sqrt{a^2-v^2}\,dv+O(\varepsilon^2).
$$
The substitution $v=a\cos\phi$ gives
$$
\int_{-a}^a\sqrt{a^2-v^2}\,dv=\frac{\pi a^2}{2},\qquad
\int_{-a}^av^2\sqrt{a^2-v^2}\,dv=\frac{\pi a^4}{8}.
$$
Consequently
$$
\Delta F=-\frac{\varepsilon\pi\sqrt3a^2}{2}(\lambda_1+a^2/4)+O(\varepsilon^2).
$$
The simple zero of this <Melnikov energy-balance method> calculation gives the <heteroclinic splitting of a fold-Hopf amplitude cycle>:
$$
\boxed{\lambda_1=\lambda_2/4+O(\varepsilon),\qquad
\mu_1=\mu_2/4+O(|\mu_2|^{3/2}),\quad\mu_2<0.}
$$
This is a leading persistence curve, not an exact parameter equality for the perturbed orbit. It lies between the two primary <Hopf bifurcation> curves near the origin and on the negative-$\mu_1$ side of the secondary torus curve $\mu_1=0$. With the added term the primary <Hopf bifurcation> curves themselves shift to $\mu_1=\pm2\sqrt{-\mu_2}+\mu_2$.
The sign of the amplitude balance also identifies the local torus branch. Over a small closed limiting contour, $\Delta F=-\varepsilon\oint u(\lambda_1+v^2)\,dv$. Its enclosed-area form sets $\lambda_1$ equal to minus the area-average of $v^2$. This average tends to zero at the centre and to $a^2/4$ at the <heteroclinic cycle>. Near the centre, negative $\lambda_1$ attracts trajectories toward the stable amplitude equilibrium; outside a zero-balance contour, $F$ decreases and the amplitude excursion grows. Thus the secondary bifurcation creates an unstable invariant <torus> on the stable-periodic-orbit side. Its continuation reaches the global connection as the balance tends to $\lambda_1=-a^2/4$. The <bifurcation diagram> distinguishes this perturbed connection from the entire degenerate family present in the unperturbed system.
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