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A global bifurcation changes an invariant set through a global orbit connection or collision that cannot be identified by a single equilibrium point's linearization alone. For example, a global bifurcation through a saddle homoclinic orbit occurs when an outgoing unstable separatrix returns to the same saddle equilibrium's stable manifold. A smooth global Poincaré return map then combines with a local passage map to determine which side contains a periodic orbit. A heteroclinic orbit instead joins distinct equilibrium points; an energy-balance or Melnikov integral can locate its breaking in a nearly Hamiltonian system.

First analyze the supplied normal form as a dynamical system in its own right. Its equilibrium points are $O=(0,0)$ and, for ${\mu }_2>0$, $S_{\pm }=(\pm \sqrt{{\mu }_2},0)$. At $O$, the trace is ${\mu }_1$ and the determinant is ${\mu }_2$. At $S_{\pm }$, the determinant is $-2{\mu }_2$, so both are saddle equilibria. Thus ${\mu }_2=0$ is the pitchfork bifurcation curve, and ${\mu }_1=0,{\mu }_2>0$ is the Hopf bifurcation curve. With the allowed supercritical Hopf bifurcation, an attracting limit cycle is born on ${\mu }_1>0$ around the repelling origin.

The Hamiltonian $H=y^2/2+{\mu }_2{x}^2/2-x^4/4$ obeys $\dot{H}=({\mu }_1-x^2)y^2$. Set $s={\mu }_2>0$, $x=\sqrt{s}X$, $y=sY$, $\tau =\sqrt{s}t$, and ${\mu }_1=cs$. The limiting Hamiltonian system has two oppositely directed heteroclinic orbits with $Y=\pm (1-X^2)/\sqrt{2}$ for $-1<X<1$. The Melnikov energy-balance method givesIts unique simple zero gives a heteroclinic cycle curveThe two connections occur on the same curve because reflection interchanges them. The attracting limit cycle grows toward this heteroclinic cycle, then disappears. In the small unfolding neighborhood there is one such connection curve, rather than two independently split curves. There are no nearby homoclinic orbits replacing it: the two equal-height outer saddles are the relevant separatrix endpoints.

This calculation transfers to the original equations only if the asserted reduction is valid. The cubic calculation in part (c) shows it is not valid for the printed coefficients. Nevertheless, the claim of a single nearby global bifurcation curve for the original equations can be proved independently. Put $\eta =-(\kappa +1)>0$, $\lambda =3+{\eta }^{2}L$, $x=\varphi =\sqrt{\eta }X$, $y=\dot{\varphi }=\eta Y$, and $\tau =\sqrt{\eta }t$. Eliminating $\theta$ using $\sin q=(y+\sin 2x)/(\cos 2x-\kappa )$ producesThe order-$\eta$ correction is reversible and has no first-order separatrix energy imbalance. On the limiting positive connection, $-1/2<X<1/2$ and $Y=1/2-2X^2$. The first dissipative balance isAgain the zero is simple, and reflection makes the two heteroclinic orbits simultaneous. Hence the actual original equations have the unique local connection curveThe unstable limit cycle on the subcritical Hopf bifurcation side reaches this curve. Thus the original system still has one nearby global connection curve, but its tangency and cycle stability differ from those of the supplied assumed cubic normal form. These statements concern connections contained in the shrinking neighborhood of $P_1$; they do not exclude unrelated distant global bifurcations elsewhere on the sphere.

For $\lambda >0$ put $s=\sqrt{\lambda }$. The equilibrium points are $(s,0)$ and $(-s,0)$, withThe positive branch is always a saddle. The negative branch is a sink for $\mu <s$ and a source for $\mu >s$; it is a node or focus according as $(\mu -s{)}^2-8s$ is positive or negative. At $\lambda =0$ the two branches meet in a saddle-node bifurcation when $\mu \ne 0$; at $(\lambda ,\mu )=(0,0)$ the double zero eigenvalue with a nontrivial Jordan block gives a Bogdanov–Takens bifurcation. There are no real equilibrium points for $\lambda <0$.

The Hopf bifurcation curve is $\mu =s>0$, equivalently $\lambda ={\mu }^2$ on its positive-$\mu$ branch. Its frequency is $\omega =\sqrt{2s}$. To determine the direction, shift $u=x+s$, $v=y$ at this threshold; thenSet $X=u$, $Y=-v/\omega$ to obtain $\dot{X}=-\omega Y$, $\dot{Y}=\omega X-X^2/\omega +XY$. A quadratic near-identity transformation, or harmonic averaging with its quadratic correction retained, gives the radial cubic coefficientwhere $g=-X^2/\omega +XY$. The radial equation is $\dot{R}=(\mu -s)R/2+{\ell }_H{R}^3+\cdots$. Hence the Hopf bifurcation is subcritical, with an unstable small periodic orbit on the stable-focus side $\mu <s$.

The saddle's outgoing separatrix can return to its incoming separatrix, creating a homoclinic orbit. This global bifurcation cannot be detected from the equilibrium point's Jacobian matrix alone. The weakly Hamiltonian scaling below locates its curve and shows how the unstable Hopf cycle grows into the saddle loop. The final diagram includes the local saddle-node and Hopf curves and this global curve; it describes the neighborhood of the double-zero point.

The rescaling resolves the small, nearby nonzero saddles while retaining the weak non-Hamiltonian terms. It allows the global stable manifold/unstable manifold connection to be studied as a perturbation of the explicit heteroclinic cycle, rather than searched for only in the original variables.

Differentiate the first integral using the first-order perturbation from part (c):On the positive unperturbed connecting branch, $u=(2s-v^2)/(2\sqrt{2})$, soIntegrating the energy change from one saddle equilibrium to the other therefore givesThis is the heteroclinic Melnikov integral for a quartic Hamiltonian. It measures the first-order energy mismatch, equivalently the transverse splitting between the outgoing unstable manifold and the incoming stable manifold. A nonzero value means that the heteroclinic orbit has broken. A simple zero supplies the leading connection condition; smooth perturbation then shifts the curve by higher-order terms. The central inversion symmetry makes the two connections have the same condition, so both can be restored on one parameter curve.

For fixed $s>0$, $\partial M/\partial \mu ={\int }_{-\sqrt{2s}}^{\sqrt{2s}}(s+v^2/2)dv>0$. The coefficient is affine in $\mu$, has a unique simple zero, and changes sign across it. No explicit evaluation is necessary. Energy drift on the nearby closed Hamiltonian levels can similarly determine a periodic orbit and its stability; the cycle born at the supercritical Hopf bifurcation grows towards the separatrix and can terminate at this global bifurcation through a heteroclinic orbit. Translating the resulting balance curve back with ${\mu }_{\mathrm{old}}={\epsilon }^2\mu$, $\sigma -1={\epsilon }^{2}s$ places it near the original double-zero point.

For $\mu >0$, the left equilibrium point $(-\sqrt{\mu },0)$ is a stable node, the right equilibrium point $(\sqrt{\mu },0)$ a saddle equilibrium, and the saddle equilibrium's left unstable branch flows directly towards the node. The right branch takes the prescribed global excursion. Its landing relative to the two equilibrium points gives the six local portraits in the figure.

If $\mu >{\nu }^2$, then $-\sqrt{\mu }<\nu <\sqrt{\mu }$ for either sign of $\nu$. The global branch lands between the equilibrium points and flows left into the node. There is no nearby Poincaré return map cycle, and the landing approaches the node from its right.

If $0<\mu <{\nu }^2$ and $\nu >0$, the branch lands to the right of the saddle equilibrium. The contracting Poincaré return map has an attracting fixed point, giving a stable periodic orbit; the node also remains as a separate attractor with its own basin. If instead $\nu <0$, the branch lands to the left of the node and approaches it from the left. There is no nearby periodic orbit in that case.

If $\mu <0$, the two equilibrium points have vanished and a stable returning cycle exists for either sign of $\nu$. For negative $\nu$ the cycle traverses the full bottleneck and has the divergent period in part (c). For fixed positive $\nu$ it enters to the right of the bottleneck; its local period approaches $1/\nu -1/h$, so it need not diverge when $\mu \to 0^{-}$. The outer arcs in the figure encode only the given return; the local equations do not specify the full vector field away from the origin.

The boundaries explain the codimension-two organization. On $\nu =\sqrt{\mu }>0$ the saddle equilibrium has a homoclinic orbit. On $\mu =0$, $\nu <0$, the return passes through the semistable equilibrium point at a saddle-node bifurcation, producing a saddle-node bifurcation on an invariant circle. On $\mu =0$, $\nu >0$, the node and saddle equilibrium instead collide away from the persistent cycle. At $\mu =\nu =0$, the returning separatrix hits the equilibrium point's strong stable manifold at the saddle-node bifurcation: this is the saddle-node separatrix-loop bifurcation. The curve $\nu =-\sqrt{\mu }$ changes which side of the node receives the returning orbit; unlike the positive branch, it is not a global bifurcation through a saddle homoclinic orbit.