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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 $a^2$.

Local equilibrium and Hopf bifurcations do not locate the larger cycles' endpoints. In the large-$s$ case, the stable origin cycle is confined between two symmetry-related saddles. A natural global completion is a heteroclinic cycle formed by the two separatrix connections between those saddles. The stable periodic orbit expands into that connecting cycle and disappears; the two connections occur together because $(q,p)\mapsto (-q,-p)$ exchanges them. This is a global transition in the positive-$\kappa$, positive-$\lambda$ region where the Bendixson-Dulac criterion permits cycles. Beyond it, the cubic inverted potential allows trajectories to leave the local region; the local reduction does not specify their ultimate fate in the original four-dimensional system.

In the small-$s$ case, the two stable cycles born around the nonzero well equilibria can encounter the central saddle simultaneously. Their limiting orbits form a symmetric figure-eight pair of homoclinic orbits. A symmetric homoclinic gluing bifurcation can then replace the two one-well cycles by one larger cycle visiting both wells. There is also an outer unstable cycle associated with the subcritical origin Hopf branch, continued into the double-well region. A saddle-node bifurcation of periodic orbits between the large stable and unstable cycles provides a possible later termination. Thus the global curves needed for the usual completion are a symmetric heteroclinic connection in the large-$s$ case, and paired homoclinic gluing together with a fold of cycles in the small-$s$ case.

These connections, and any resulting basin changes, are not detected by the equilibrium Jacobians. Their precise parameter curves require a global return-map or separatrix calculation, which is not part of this requested qualitative description. Symmetry forces paired connections where indicated, but the local centre manifold and Hopf calculations alone should not be presented as numerical locations or a proof of the full large-amplitude diagram.

The polar equations areEvery positive zero of the radial factor is therefore a circular periodic orbit of period $2\pi$. Writing $s=r^2$, the equation isThe resulting count isFor the two-orbit range, $s_{-}<\lambda /2<s_{+}$. Differentiating the radial vector field at an orbit givesso the inner orbit is unstable and the outer orbit is stable. Atthey meet in a saddle-node bifurcation of periodic orbits. Atthe inner orbit shrinks into the origin, giving the second bifurcation.

Put $x=z$, $y=\dot{z}$. The system isIts equilibria are $\boxed{(0,0),(1,0),(-1,0)}$. At $(\pm 1,0)$, the eigenvalues are $(-A\pm \sqrt{A^2+8})/2$, of opposite signs, so these are saddles for every $A$.

At the origin, the eigenvalues are $(-A\pm \sqrt{A^2-4})/2$. It is a stable focus for $0<A<2$, a stable node for $A>2$, an unstable focus for $-2<A<0$, and an unstable node for $A<-2$. At $A=\pm 2$ it is the corresponding repeated-eigenvalue improper node. At $A=0$, linearization gives a center, but nonlinear damping matters: withsmall closed energy sublevels are invariant and $\dot{E}=-y^4$ forces convergence to the origin. Thus it is a weak attracting focus, not a nonlinear center.

Local bifurcation. Write $a=-A$. Averaging the radial equation near the origin, or eliminating nonresonant cubic terms, givesThe $x^3$ restoring term has zero average radial contribution, while $⟨y^4⟩=3R^4/8$. Thus $A=0$ is a supercritical Hopf bifurcation as $a$ increases: a stable cycle appears for small $a>0$, with leading amplitude $R\simeq \sqrt{4a/3}$.

Global changes. Two further bifurcations must be retained. Define $a_h$ as the parameter where the upper unstable branch from $(-1,0)$ reaches $(1,0)$. Central symmetry supplies the lower connection in the opposite direction, forming a heteroclinic cycle. Define $a_f>a_h$ as the subsequent collision of an inner attracting and an outer repelling periodic orbit. The complete phase portraits for $|A|<1$ are:

Here escape means leaving the central well and becoming unbounded; it is not an additional finite equilibrium. The softening Rayleigh-Duffing phase portrait in the figure includes the two critical cases.

The return-map calculation underlying this classification is useful because the two global critical parameters are very close. On the section $x=0,y>0$, let $H_a(p)$ be minus the velocity at the next crossing of $x=0$, so a symmetric cycle is a root of $H_a(p)-p$. Any periodic orbit must have its extrema inside $-1<x<1$, since at a positive maximum $y=0$ requires $x^3-x<0$. Its index is one, so it surrounds the origin and neither saddle. Every such cycle is centrally symmetric: its reflected image cannot be a distinct nonintersecting cycle of equal enclosed area. Thus this half-return test detects all central cycles. A fixed point with $H_a^{\prime }<1$ is attracting and one with $H_a^{\prime }>1$ repelling; the full-return multiplier is $(H_a^{\prime }{)}^2$.

For the saddle connection, let $p_L(a)$ be the upper velocity at $x=0$ from the left saddle's unstable branch and $p_R(a)$ the upper velocity at $x=0$ on the right saddle's stable branch. While $y>0$, their shooting equation isComparison shows $p_L$ increases with $a$ and $p_R$ decreases. At $a=0$, energy dissipation gives $p_L<1/\sqrt{2}<p_R$. At $a=1$, the lower comparison curve $y=(x+1)/2$ on $[-1,0]$ gives $p_L>1/2$, while the upper comparison curve $y=(1-x^2)/2$ on $[0,1]$ gives $p_R<1/2$. The latter follows fromand the stable-branch expansion $y=(1-x)-3(1-x{)}^2/4+\cdots$ near $x=1$. Hence there is a unique $a_h\in (0,1)$ with $p_L=p_R$.

At that connection the saddle eigenvalues are ${\lambda }_u=(a_h+\sqrt{a_h^2+8})/2$ and ${\lambda }_s=(a_h-\sqrt{a_h^2+8})/2$, so $|{\lambda }_s|/{\lambda }_u<1$. The loop is therefore repelling. In particular, the attracting Hopf cycle cannot simply disappear in that connection. Near the upper endpoint $p_R$, the half-return map hasFor $a$ just above $a_h$, this produces a second, repelling fixed point near the endpoint, in addition to the attracting inner one. Continuing the two roots of $H_a(p)-p$ makes their collision criterion explicit:This is a saddle-node bifurcation of periodic orbits, not a disappearance of the equilibrium points.

For a reproducible numerical check of the global sketches, event-based integration of this half-return map and its variational equation givesThese values are computed by shooting the stated equations; their precision is unnecessary for the requested sketches. They explain why a coarse parameter sweep can miss the two-cycle interval entirely. The three bifurcation types are therefore Hopf at $A=0$, a two-saddle heteroclinic connection at $A=-a_h$, and a fold of cycles at $A=-a_f$. There is no pitchfork of equilibria because their locations remain fixed for every $A$.

The softening restoring force and cubic velocity damping give the energy $E=(z^{\prime }{)}^2/2+z^2/2-z^4/4$ with $\dot{E}=-A(z^{\prime }{)}^2-(z^{\prime }{)}^4$. The origin and the two saddle points $z=\pm 1$ persist for all parameters. The origin undergoes a supercritical Hopf bifurcation at $A=0$. A heteroclinic cycle and a subsequent saddle-node bifurcation of periodic orbits are distinct global changes in its softening Rayleigh-Duffing phase portrait.

With $a=-A>0$, a stable cycle grows from the origin. At $a_h\simeq 0.33144619$, symmetric saddle connections form a repelling heteroclinic cycle, while the stable inner cycle persists. Just above it an unstable outer cycle coexists with the stable one; they merge at $a_f\simeq 0.33147066$ in a saddle-node bifurcation of periodic orbits. These approximate values follow from the half-return equations $H_a(p)=p$, with $H_a^{\prime }(p)=1$ at the fold, and saddle-manifold shooting at the connection. A coarse parameter scan can miss the narrow coexistence interval. The saddle eigenvalue ratio is below one at the connection, proving that the loop is repelling and preventing identification of the stable cycle's disappearance with that connection.