= Solution
A persistent saddle loop must return with the same energy. The <Melnikov energy-balance method> therefore requires, at first order,
$$
0=\varepsilon\int_{-\infty}^{\infty}(\beta+u_0)v_0^2\,d\tau+O(\varepsilon^2).
$$
For the normalized orbit of part (c), put $z=\tanh(\tau/\sqrt2)$. Since $u_0=-2+3z^2$, $v_0^2=18(1-z^2)^2z^2$ and $d\tau=\sqrt2\,dz/(1-z^2)$,
$$
\begin{aligned}
I_0&=18\sqrt2\int_{-1}^{1}(z^2-z^4)\,dz=\frac{24\sqrt2}{5},\\
I_1&=18\sqrt2\int_{-1}^{1}(-2+3z^2)(z^2-z^4)\,dz=-\frac{24\sqrt2}{7}.
\end{aligned}
$$
For $s=\sqrt\alpha$ the general orbit has $I_0(s)=s^{5/2}I_0$ and $I_1(s)=s^{7/2}I_1$. The energy mismatch is therefore proportional to $I_0(s)[\beta-(5/7)s]$, whose derivative with respect to $\beta$ is nonzero. This simple zero selects the saddle-loop curve:
$$
\boxed{\beta=\frac57\sqrt\alpha+O(\varepsilon),\qquad
\mu_h=\frac57\sqrt\lambda+O(\lambda^{3/4}).}
$$
The remainder can be sharpened. Reversal of $\tau$ and $v$ interchanges $\varepsilon$ and $-\varepsilon$ without changing the saddle-loop condition. Smoothness of the homoclinic splitting thus makes the selected $\beta$ an even function of $\varepsilon$, giving
$$
\boxed{\mu_h(\lambda)=\frac57\sqrt\lambda+O(\lambda).}
$$
The leading coefficient alone is enough to separate it from the Hopf curve $\mu_H=\sqrt\lambda$. The integral hint in the PDF equals $2/(k+1)$ for nonnegative even integers $k$; for odd $k$ its integrand is odd and the integral is zero. Only even powers enter the calculation above.
Near the double-zero point, and with the stipulated absence of further bifurcations, the distinct open regions are:
* $\lambda<0$: no <equilibrium points> or local <periodic orbits>; $\dot y<0$ wherever $y=0$, and the flow passes through the bottleneck region.
* $\lambda>0$, $\mu<\mu_h$: a saddle and a stable node or focus, with no local <periodic orbit>.
* $\lambda>0$, $\mu_h<\mu<\sqrt\lambda$: a saddle, a stable focus and an unstable <periodic orbit> surrounding the focus. This cycle is its local basin boundary.
* $\lambda>0$, $\mu>\sqrt\lambda$: a saddle and an unstable node or focus, with no small <periodic orbit>.
On the homoclinic curve the cycle becomes a saddle loop; its positive saddle <trace> $\mu+\sqrt\lambda$ makes the loop repelling on the cycle side. At the Hopf curve the unstable cycle shrinks into the negative <equilibrium point>. The node–focus <discriminant> curves merely change the spiral appearance and are not additional bifurcations. The impossibility of a <periodic orbit> for $\lambda<0$ can also be seen without a picture: at a local minimum of periodic $x(t)$ one would have $y=0$ and $\ddot x=-\lambda+x^2>0$, but at its maximum the same strictly positive expression would be required to be nonpositive.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-65-bogdanov-takens.png]
{title=Local saddle-node, subcritical Hopf and homoclinic curves with the four nearby phase portraits}
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