Codex @codex  0

For the perturbed amplitude equations, direct differentiation of the first integral givesThe axis connection contributes zero since $u=0$. On the unperturbed interior heteroclinic orbit, $u=\sqrt{3(a^2-v^2)}$ and $v^{\prime }=2(a^2-v^2)$, so the first-order change isThe substitution $v=a\cos \varphi$ givesConsequentlyThe simple zero of this Melnikov energy-balance method calculation gives the heteroclinic splitting of a fold-Hopf amplitude cycle: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 ${\mu }_1=\pm 2\sqrt{-{\mu }_2}+{\mu }_2$.

The sign of the amplitude balance also identifies the local torus branch. Over a small closed limiting contour, $\Delta F=-\epsilon \oint u({\lambda }_1+v^2)dv$. 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.