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Perturb the first integral of the quadratic fold-Hopf amplitude flow by $u^{\prime }=-2uv+\epsilon u({\lambda }_1+v^2)$, keeping $v^{\prime }={\lambda }_2+v^2+u^2$. Then $F^{\prime }=-\epsilon u({\lambda }_1+v^2)({\lambda }_2+u^2+v^2)$. On the interior heteroclinic orbit for ${\lambda }_2=-a^2$, $u=\sqrt{3(a^2-v^2)}$ and $v^{\prime }=2(a^2-v^2)$, so the first-order change is $-\epsilon \sqrt{3}{\int }_{-a}^a({\lambda }_1+v^2)\sqrt{a^2-v^2}dv=-\epsilon \pi \sqrt{3}{a}^2({\lambda }_1+a^2/4)/2$. The boundary connection contributes zero. The simple zero gives the displayed leading persistence curve, rather than an exact all-orders parameter equality.

Within the positive-amplitude wedge the reduced periodic orbit equilibrium has amplitude trace ${\mu }_1$ and positive determinant $4r_{\ast }^2$. At ${\mu }_1=0,{\mu }_2<0$ the transverse pair is imaginary, so the possible secondary event is a Hopf bifurcation of the amplitude equations, corresponding to an invariant torus in the full system.

It is degenerate. Set ${\mu }_2=-a^2$ and ${\mu }_1=0$. The first integral of the quadratic fold-Hopf amplitude flow isThe amplitude equations have the Hamiltonian system form $\dot{r}=F_x$, $\dot{x}=-F_r$, so $\dot{F}=0$. At $(r,x)=(a,0)$, $F$ has a strict local maximum $2a^3/3$ and negative-definite Hessian. Every nearby regular level is a closed amplitude periodic orbit, rather than one isolated orbit selected by nonlinear damping. Adding the rotating phase gives an entire family of invariant tori; the full motion is quasiperiodic or periodic according to the phase advance during one amplitude period.

For ${\mu }_1\ne 0$, the amplitude vector field has constant divergence ${\mu }_1$. The Bendixson-Dulac theorem rules out closed amplitude trajectories in the physical simply connected half-plane $r>0$. Hence there is no ordinary branch of isolated invariant tori on either side in this quadratic system. The secondary event has vanishing radial saturation coefficients; its imaginary linear pair alone does not establish a nondegenerate torus bifurcation.