= Heteroclinic splitting of a fold-Hopf amplitude cycle
{title2=$\lambda_1=\lambda_2/4+O(\varepsilon)$}
Perturb the <first integral of the quadratic fold-Hopf amplitude flow> by $u'=-2uv+\varepsilon u(\lambda_1+v^2)$, keeping $v'=\lambda_2+v^2+u^2$. Then $F'=-\varepsilon u(\lambda_1+v^2)(\lambda_2+u^2+v^2)$. On the interior <heteroclinic orbit> for $\lambda_2=-a^2$, $u=\sqrt{3(a^2-v^2)}$ and $v'=2(a^2-v^2)$, so the first-order change is $-\varepsilon\sqrt3\int_{-a}^a(\lambda_1+v^2)\sqrt{a^2-v^2}\,dv=-\varepsilon\pi\sqrt3\,a^2(\lambda_1+a^2/4)/2$. The boundary connection contributes zero. The simple zero gives the displayed leading persistence curve, rather than an exact all-orders parameter equality.
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