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Near two homoclinic orbits of a planar saddle equilibrium with contracting ratio $\delta >1$, the signed Poincaré return map has separate offsets $\mu ,\nu$. Two single-lobe periodic orbits exist for $\mu <0$ and $\nu <0$, respectively. A two-lobe periodic orbit is represented by positive $p,q$ solving the displayed equations. Its boundary consists of the compound homoclinic orbit curves $\mu =-A{\nu }^{\delta }$ for $\nu >0$ and $\nu =-B{\mu }^{\delta }$ for $\mu >0$, with $A,B>0$. In a sufficiently small return domain the derivatives tend to zero, so each admissible symbolic branch has at most one attracting periodic orbit. Restricting to $\mu =\nu$, $A=B$ gives the usual symmetric homoclinic gluing bifurcation.

A Bogdanov–Takens bifurcation has a double zero eigenvalue with a nontrivial Jordan block and generic two-parameter unfolding. Its local diagram includes saddle-node bifurcations, Hopf bifurcations, and a curve of small homoclinic orbits. A weakly perturbed Hamiltonian scaling can locate the homoclinic curve through energy balance.

For unit-mass motion in $V(x)=3x^2-2x^3$, the mechanical energy is $E=v^2/2+V(x)$. The origin is a center equilibrium, and $(1,0)$ is a saddle equilibrium. Energies $0<E<1$ have trapped periodic orbits around the origin as well as a separate escaping component to the right. The barrier energy has a homoclinic orbit returning to the saddle after a turn at $x=-1/2$, plus two saddle branches on $x>1$. Above the barrier, particles traverse the well and eventually escape to the right.

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.

For $\dot{r}=r(1-r)(r-2\sin \theta )$, $\dot{\theta }=r$, the circle $r=1$ has normal Floquet multiplier $e^{-2\pi }$ and is attracting. Inside it, $w=(1-r{)}^{-1}$ satisfies $dw/d\theta =(1-2\sin \theta )w-1$. The solution through $w(\pi )=1$ has $w>1$ on one interval $({\theta }_b,\pi )$ with $-\pi <{\theta }_b<0$. The corresponding radius $r=1-1/w$ is a homoclinic orbit to the origin. Near its returning endpoint $r\sim (\theta -\pi {)}^2$, so the approach takes infinite physical time. Comparison for the scalar angular equation shows that smaller radii inside this loop approach the origin, whereas all positive radii outside it approach the circle. The origin is therefore not Lyapunov stable despite attracting an open set.

The conservative equation $u^{\prime \prime }=u^2-a^2$ has a saddle loop with $v_{+}(u)=\sqrt{2/3}(a-u)\sqrt{u+2a}$ for $-2a\le u\le a$. Its homoclinic orbit integrals are $I_0=2{\int }_{-2a}^a{v}_{+}(u)du$ and $I_1=2{\int }_{-2a}^{a}u{v}_{+}(u)du$. Substitution $s=u+2a$ evaluates them as displayed. For a weak perturbation $\epsilon (\beta +u)u^{\prime }$, integrating the energy derivative gives $\beta I_0+I_1=0$, hence $\beta =5a/7$. The nonzero derivative $I_0$ makes this a simple first-order splitting zero.

The equilibria are $0$ and $-\epsilon \pm \sqrt{{\epsilon }^2+\lambda }$. Their linear trace is $\kappa -v_{\ast }^2$ and determinant $\lambda -4\epsilon v_{\ast }-3v_{\ast }^2$. For nonzero $\epsilon$ the symmetry-protected pitchfork bifurcation splits into a transcritical bifurcation at $\lambda =0$ and a saddle-node bifurcation at $\lambda =-{\epsilon }^2$. The energy $H={\dot{v}}^2/2+\lambda v^2/2-2\epsilon v^3/3-v^4/4$ obeys $\dot{H}=(\kappa -v^2){\dot{v}}^2$. For positive $\lambda$, the two outer saddle equilibria have unequal barrier heights, so the symmetric heteroclinic cycle unfolds into separated one-way connections and a homoclinic orbit termination of the central stable limit cycle.

At a saddle-focus homoclinic orbit, the leading Shilnikov return map is $f(x)=A{x}^{\delta }\cos (q\log x+\Phi )$, $x>0$, with $q>0$. If $1/2<\delta <1$, its fixed points solve $\cos (q\log x+\Phi )=x^{1-\delta }/A$. Infinitely many solutions accumulate at zero near the zeros of the cosine, with successive size ratios tending to $e^{-\pi /q}$. The derivative is $A{x}^{\delta -1}[\delta \cos (q\log x+\Phi )-q\sin (q\log x+\Phi )]$, so their expanding multiplier is unbounded. Narrow monotone strips near successive oscillations map across a common interval; inverse branches contract, supplying arbitrarily long symbolic itineraries and infinitely many saddle periodic orbits. Varying the splitting parameter shifts the graph vertically, producing accumulating fold and period-doubling thresholds and stable windows. The condition $\delta >1/2$ makes the local flow volume-contracting, whereas $\delta <1$ makes its saddle value positive.

For $u^{\prime }=v$, $v^{\prime }=u-u^3+\epsilon (\beta -u^2)v$, let $H=v^2/2-u^2/2+u^4/4$. On an outer periodic orbit with $H>0$, leading energy balance method requires $\beta =B(H)$, whereHere $u_{\max }^2=1+\sqrt{1+4H}$. The homoclinic orbit limit is $B(0)=4/5$. Differentiating the denominator gives a period integral diverging at $H=0$, whereas the derivative of the numerator stays finite; hence $B$ decreases immediately above zero. At large $H$, rescaling $u=H^{1/4}s$ gives $B(H)\propto \sqrt{H}$. Its first nondegenerate minimum selects a saddle-node bifurcation of periodic orbits; numerical quadrature gives ${\beta }_{\mathrm{fold}}\simeq 0.75226$. The cycle on the decreasing portion is unstable and the one on the increasing portion is stable, since the net energy drift has sign $\beta -B(H)$. Restoring an unperturbed coefficient $a^2$ in the linear restoring term multiplies the threshold by $a^2$.

Substitution of the weighted scaling and $\tau =\epsilon t$ givesThe derivative of the candidate Hamiltonian isThus $H$ is a conserved quantity at $\epsilon =0$. For $\alpha >0$ put $a=\sqrt{\alpha }$. The conservative equilibrium point $(-a,0)$ is a center with $H=-2a^3/3$, and $(a,0)$ is a saddle equilibrium with saddle-loop energyThe closed component of a level between these energies surrounds the center. At $H=H_s$,is the homoclinic orbit, with left turning point $u=-2a$. The upper branch is traversed to the right and the lower branch to the left. Levels above the barrier have escaping trajectories; the saddle level also has unbounded branches to the right. For $\alpha =0$ the two critical points coalesce; for $\alpha <0$ the potential is monotone and there is no center or saddle loop.

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.

Introduce $H=x^2-y^2+y^4/2$. Along a solution,At $\alpha =0$, $H$ is a first integral. The identity $H+1/2=x^2+(y^2-1{)}^2/2$ shows its level sets are bounded: neither $x$ nor $y$ can escape to infinity on a fixed level. Consequently trajectories remain bounded as $t\to \infty$, including those starting far from the origin. This bound also prevents finite-time escape; no explicit solution is needed.

The equilibrium points are $(0,0)$ and $(0,\pm 1)$. The Jacobian matrix at $(0,y_{\ast })$ isAt the origin its eigenvalues are $(\alpha \pm \sqrt{{\alpha }^2+4})/2$, with opposite signs, so the origin is a saddle point in all three cases. An eigenvector for $\lambda$ lies on $y=-x/\lambda$; the stable direction has positive slope and the unstable direction negative slope. At $(0,\pm 1)$ the eigenvalues are $(\alpha \pm \sqrt{{\alpha }^2-8})/2$.

When $\alpha =0$, these latter eigenvalues are $\pm i\sqrt{2}$. The nonlinear classification follows from the strict minima of $H$ at the two points: nearby level sets are closed curves without other equilibria, and the nonvanishing vector field makes them periodic orbits. Thus the two points are center equilibria. For $-1/2<H<0$, there are two separate closed ovals, one around each center equilibrium. The level $H=0$ consists of the saddle point and two homoclinic orbits forming a figure eight, explicitlyFor $H>0$, a single outer closed orbit encloses all three equilibrium points. Motion is clockwise, as the vector points downwards wherever $x>0$. The upper homoclinic orbit leaves the origin into $x<0,y>0$ and returns through $x>0,y>0$; the lower one has the reflected orientation.

When $\alpha =0.1$, the two nonzero equilibrium points have eigenvalues $0.05\pm i\sqrt{7.99}/2$ and are unstable foci. Nearby paths spiral outward clockwise. The first integral becomes a strictly increasing energy along every nonconstant trajectory: although $\dot{H}$ can momentarily vanish when $x=0$, it cannot vanish on an interval unless the solution is an equilibrium. Thus there are no periodic orbits or homoclinic orbits. The two stable separatrices of the saddle point, traced backwards, spiral towards the corresponding unstable foci. Its unstable separatrices move onto positive energy and escape outwards. All other nonequilibrium trajectories outside the stable manifold are unbounded forwards: a bounded forward limit set would have to lie in $x=0$, whose only invariant points are the three equilibria; the two unstable foci cannot attract a nonconstant trajectory, and convergence to the saddle occurs only on its stable manifold. Escape is not finite-time blow-up, since $\dot{H}\le 2\alpha (H+1/2)$ bounds the energy on every finite interval.

When $\alpha =-0.1$, the eigenvalues at the nonzero points are $-0.05\pm i\sqrt{7.99}/2$, so they are stable foci. The energy decreases, keeping all forward trajectories in compact sublevel sets. A bounded limit set must lie in the invariant part of $x=0$, so trajectories converge to equilibria: generically to one of the two stable foci, exceptionally to the saddle along its stable manifold. The saddle's unstable separatrices spiral into the two stable foci; its stable manifold separates their attraction basins. Again, energy monotonicity excludes periodic orbits and homoclinic orbits. These three cases give the quartic double-well phase portrait with linear damping below.

Here $\mu >{\mu }_H=\lambda (\lambda -1)>0$. The origin is a saddle equilibrium, $P_{+}$ is repelling, and $P_{-}$ lies in $u<0$. Thus there is no saddle equilibrium in the interior of $u>0$ to which this periodic orbit could become homoclinic. The origin cannot supply the missing loop: the line $u=0$ is invariant and is precisely its local stable manifold. An orbit in $u>0$ cannot lie on that stable manifold, so the positive periodic orbit cannot close into a homoclinic orbit to the origin either.

The outgoing trajectories do not escape to infinity in this regime. To see the large-amplitude turning mechanism, put $w=v/u$ once $v\ge 0$. Direct differentiation givesSince $\lambda >1$, the quadratic $w^2-w+\lambda$ is uniformly positive. At large $u$ while $0\le w\le 1$, $\dot{w}$ is bounded below by a positive multiple of $u$, whereas $\dot{u}/u\le \mu +u$. Thus $w$ rises by a fixed amount per bounded increase in $\log u$: the trajectory reaches $w>1$ and then $w>1+\mu /u$, where $\dot{u}<0$, before $u$ can diverge. The same estimate bounds repeated outgoing excursions; large positive $v$ also decreases when $v>u^2$. Initially negative $v$ increases until it becomes nonnegative. These nullcline and ratio estimates give bounded forward excursions.

The repelling upper equilibrium is therefore surrounded by recurrent trajectories, and the Poincaré-Bendixson alternative supplies periodic motion rather than a saddle equilibrium connection. The supercritical attracting periodic orbit persists in the displayed regime, with trajectories spiralling away from the upper equilibrium and approaching it. The last panel of the portrait figure shows this behavior; the boundary stable manifold of the origin is kept separate.

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 supplied normal form, the four local topological phase portraits are as follows. When ${\mu }_2<0$, only the origin is present and it is a saddle equilibrium. When ${\mu }_2>0,{\mu }_1<0$, the origin is attracting, between the two outer saddle equilibria; inward-facing separatrices enter its basin. When ${\mu }_2>0$ and $0<{\mu }_1<h({\mu }_2)$, the origin is repelling and a stable limit cycle surrounds it, inside the separatrix boundary formed by the outer saddle equilibria. Finally, when ${\mu }_2>0,{\mu }_1>h({\mu }_2)$, the origin remains repelling but the local limit cycle has disappeared, and trajectories leave the central region through the opened separatrix channels. Whether the origin is a node or a focus within one of these regions does not change this orbit topology.

On the connection curve, the two outer saddle equilibria are joined in both directions; the period of the disappearing limit cycle diverges. This is a heteroclinic cycle, not a pair of saddle homoclinic orbits. The sketches below show the four open regions of the supplied system; they are not presented as phase portraits of the inequivalent printed spherical equations.

We count periodic orbits whose trajectories remain near the two specified homoclinic orbits; the local hypotheses impose no bound on unrelated distant cycles. Choose the return domain small enough that $A\delta |x{|}^{\delta -1}$ and $B\delta |x{|}^{\delta -1}$ are less than one. Each admissible return branch is then a contraction mapping.

A right-lobe periodic orbit is a fixed point $(p,+1)$ with $p>0$ and $p=-\mu +A{p}^{\delta }$. For small amplitudes it exists exactly when $\mu <0$, and $p=-\mu +o(|\mu |)$. Similarly, a left-lobe periodic orbit has $(x,y)=(-q,-1)$, $q>0$, $q=-\nu +B{q}^{\delta }$, and exists for $\nu <0$. Their Floquet multipliers tend to zero, so both are attracting.

A period-two orbit of the section map corresponds to one flow periodic orbit traversing both lobes. Write its states as $(p,-1)$ and $(-q,+1)$. Its positive amplitudes satisfyThe two-step derivative is $AB{\delta }^2{p}^{\delta -1}{q}^{\delta -1}$, again tending to zero. The asymmetric planar gluing return map therefore gives at most one such attracting two-lobe cycle in the small domain. Its boundaries are obtained by setting one amplitude to zero:For the original flow these are leading equations, with corrections $o({\nu }^{\delta })$ or $o({\mu }^{\delta })$. On $C_R$, the negative unstable branch makes both excursions before returning to the positive stable branch; on $C_L$ the positive unstable branch returns to the negative stable branch. Each is a compound homoclinic orbit, with one extra passage near the saddle equilibrium. They are distinct from the primary loop curves $\mu =0$ and $\nu =0$.

The six open regions and their local cycles are:

There is no additional small alternating cycle when both parameters are negative. Indeed the two positive-amplitude equations would imply $q<A{p}^{\delta }$ and $p<B{q}^{\delta }$, giving $p<B{A}^{\delta }{p}^{{\delta }^2}$, impossible for sufficiently small positive $p$. On each mixed-sign quadrant the contraction equations give the two-lobe cycle precisely on the side of the compound curve toward the positive quadrant.

On $\mu =0$, there is the primary right homoclinic orbit, together with a left cycle if $\nu <0$ or a two-lobe cycle if $\nu >0$. On $\nu =0$ the analogous statement holds with left and right interchanged. On $C_R$ there is a compound homoclinic orbit and the right single-lobe cycle; on $C_L$ there is a compound homoclinic orbit and the left single-lobe cycle. At $(0,0)$ the two primary loops meet at the same saddle equilibrium. The approach to each homoclinic orbit has a diverging period; it is not a saddle-node collision of two finite-period cycles.