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For small $\epsilon$, the conserved quantity from part (b) instead obeys $H^{\prime }=-\epsilon (\lambda +u)v^2$. The Melnikov energy-balance method evaluates the first-order net change along the unperturbed homoclinic orbit:Using the homoclinic curve $v^2=\frac{2}{3}(s-u{)}^2(u+2s)$, these are integrals of $vdu$ on its two halves. Their weighted mean is $I_1/I_0=-5s/7$. Thus the first-order splitting vanishes at $\lambda =5s/7$. Since its derivative with respect to $\lambda$ is $-I_0\ne 0$, the implicit function theorem gives a nearby curve where the stable and unstable saddle separatrices reconnect. This is the homoclinic balance for a quadratic-force oscillator and therefore a global, rather than local, bifurcation.

Returning to the original parameters, $q={\epsilon }^2\lambda$ and $\sqrt{p}={\epsilon }^2\sqrt{\kappa }$. With fixed positive $\kappa$ and $\epsilon \to 0$, the homoclinic bifurcation curve has leading formThe energy integral locates the leading curve; the perturbed orbit itself is no longer a constant-$H$ curve.