= Homoclinic integrals for a quadratic-force oscillator
{title2=$I_0={24\sqrt2\over5}a^{5/2},\quad I_1=-{24\sqrt2\over7}a^{7/2}$}
The conservative equation $u''=u^2-a^2$ has a saddle loop with $v_+(u)=\sqrt{2/3}(a-u)\sqrt{u+2a}$ for $-2a\le u\le a$. Its <homoclinic orbit> integrals are $I_0=2\int_{-2a}^av_+(u)du$ and $I_1=2\int_{-2a}^au v_+(u)du$. Substitution $s=u+2a$ evaluates them as displayed. For a weak perturbation $\varepsilon(\beta+u)u'$, integrating the energy derivative gives $\beta I_0+I_1=0$, hence $\beta=5a/7$. The nonzero derivative $I_0$ makes this a simple first-order splitting zero.
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