 Homoclinic balance for a quadratic-force oscillator

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.