Homoclinic balance for a quadratic-force oscillator ($\lambda =\frac{5}{7}\sqrt{\kappa }$)

For $u^{\prime \prime }=u^2-\kappa -\epsilon (\lambda +u)u^{\prime }$ with $\kappa >0$, the Hamiltonian homoclinic orbit has $H=2{\kappa }^{3/2}/3$. Its energy balance is $\lambda \int v^{2}dt+\int u{v}^{2}dt=0$. Along this orbit the ratio of the second integral to the first is $-5\sqrt{\kappa }/7$, giving the displayed simple zero and hence the leading homoclinic parameter curve.

The conservative equation $u^{\prime \prime }=u^2-a^2$ 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 $I_0=2{\int }_{-2a}^a{v}_{+}(u)du$ and $I_1=2{\int }_{-2a}^{a}u{v}_{+}(u)du$. Substitution $s=u+2a$ evaluates them as displayed. For a weak perturbation $\epsilon (\beta +u)u^{\prime }$, 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.