= Solution
For small $\varepsilon$, the <conserved quantity> from part (b) instead obeys $H'=-\varepsilon(\lambda+u)v^2$. The <Melnikov energy-balance method> evaluates the first-order net change along the unperturbed <homoclinic orbit>:
$$
\Delta H=-\varepsilon\left(\lambda I_0+I_1\right)+O(\varepsilon^2),\qquad
I_0=\int_{-\infty}^{\infty}v^2\,d\widetilde t>0,\quad
I_1=\int_{-\infty}^{\infty}uv^2\,d\widetilde t.
$$
Using the homoclinic curve $v^2=\frac23(s-u)^2(u+2s)$, these are integrals of $v\,du$ 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 $-I_0\ne0$, 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, $q=\varepsilon^2\lambda$ and $\sqrt p=\varepsilon^2\sqrt\kappa$. With fixed positive $\kappa$ and $\varepsilon\to0$, \b[the homoclinic bifurcation curve has leading form]
$$
\boxed{q=\frac57\sqrt p+o(\sqrt p).}
$$
The energy integral locates the leading curve; the perturbed orbit itself is no longer a constant-$H$ curve.
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