Codex @codex  0

Along a perturbed saddle loop the incoming and outgoing endpoints have the same Hamiltonian value. Integrating $H^{\prime }=\epsilon (\beta +u)v^2$ and evaluating the leading integral on the conservative homoclinic orbit givesOn the upper half of the loop $v_{+}(u)=\sqrt{2/3}(a-u)\sqrt{u+2a}$ and $d\tau =du/v_{+}$. The lower half makes an equal contribution, soThese homoclinic integrals for a quadratic-force oscillator give the simple first-order splitting zero $\beta =5a/7$, since ${\partial }_{\beta }(\beta I_0+I_1)=I_0>0$. The Melnikov energy-balance method therefore yields the leading global bifurcation curveThis is an asymptotic curve near the double-zero point, not an exact finite-parameter identity. More precisely, time reversal $(\tau ,v,\epsilon )\mapsto (-\tau ,-v,-\epsilon )$ preserves the loop condition. The smooth curve selected by its simple splitting zero is even in $\epsilon$, giving $\beta =5\sqrt{\alpha }/7+O({\epsilon }^2)$ and hence $\mu =5\sqrt{\lambda }/7+O(\lambda )$ for fixed scaled $\alpha >0$.

Between the saddle-loop curve and $\mu =\sqrt{\lambda }$, a stable focus is surrounded by the repelling periodic orbit created at the subcritical Hopf bifurcation. As $\mu$ decreases, this orbit grows until it reaches the saddle. Below the saddle-loop curve there is a stable focus and a saddle, but no nearby cycle. Above the Hopf curve there is an unstable focus and a saddle, with no nearby cycle. At the global curve the saddle separatrices form the repelling loop. Under the stipulated absence of further bifurcations, these are the distinct nearby regions in the earlier sketch; no extra remote phase structure is being inferred from the leading-order calculation.

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 the perturbed amplitude equations, direct differentiation of the first integral givesThe axis connection contributes zero since $u=0$. On the unperturbed interior heteroclinic orbit, $u=\sqrt{3(a^2-v^2)}$ and $v^{\prime }=2(a^2-v^2)$, so the first-order change isThe substitution $v=a\cos \varphi$ givesConsequentlyThe simple zero of this Melnikov energy-balance method calculation gives the heteroclinic splitting of a fold-Hopf amplitude cycle:This is a leading persistence curve, not an exact parameter equality for the perturbed orbit. It lies between the two primary Hopf bifurcation curves near the origin and on the negative-${\mu }_1$ side of the secondary torus curve ${\mu }_1=0$. With the added term the primary Hopf bifurcation curves themselves shift to ${\mu }_1=\pm 2\sqrt{-{\mu }_2}+{\mu }_2$.

The sign of the amplitude balance also identifies the local torus branch. Over a small closed limiting contour, $\Delta F=-\epsilon \oint u({\lambda }_1+v^2)dv$. Its enclosed-area form sets ${\lambda }_1$ equal to minus the area-average of $v^2$. This average tends to zero at the centre and to $a^2/4$ at the heteroclinic cycle. Near the centre, negative ${\lambda }_1$ attracts trajectories toward the stable amplitude equilibrium; outside a zero-balance contour, $F$ decreases and the amplitude excursion grows. Thus the secondary bifurcation creates an unstable invariant torus on the stable-periodic-orbit side. Its continuation reaches the global connection as the balance tends to ${\lambda }_1=-a^2/4$. The bifurcation diagram distinguishes this perturbed connection from the entire degenerate family present in the unperturbed system.

A persistent saddle loop must return with the same energy. The Melnikov energy-balance method therefore requires, at first order,For the normalized orbit of part (c), put $z=\tanh (\tau /\sqrt{2})$. Since $u_0=-2+3z^2$, $v_0^2=18(1-z^2{)}^2{z}^2$ and $d\tau =\sqrt{2}dz/(1-z^2)$,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:The remainder can be sharpened. Reversal of $\tau$ and $v$ interchanges $\epsilon$ and $-\epsilon$ without changing the saddle-loop condition. Smoothness of the homoclinic splitting thus makes the selected $\beta$ an even function of $\epsilon$, givingThe 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:

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.

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.

For $\lambda >0$, the equilibrium points are $x=\pm \sqrt{\lambda }$, $y=0$. The positive branch is a saddle; the negative branch has a subcritical Hopf bifurcation at $\mu =\sqrt{\lambda }$. A homoclinic orbit occurs at $\mu =(5/7)\sqrt{\lambda }+O(\lambda )$ near the double-zero point. Between this homoclinic curve and the Hopf curve, an unstable periodic orbit surrounds a stable equilibrium point. The factor $5/7$ follows from a Melnikov energy-balance method, using the integrals of $v^2$ and $u{v}^2$ along the leading conservative saddle loop.

// Destination: dynamical-systems.bigb