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The equilibrium points satisfy $y=0$, $x^2=\lambda$. For $\lambda <0$ there are none. For $\lambda >0$ set $a=\sqrt{\lambda }$. The Jacobian matrix isThe positive equilibrium point $(a,0)$ is always a saddle equilibrium. At $(-a,0)$ the determinant is $2a$, so it is stable for $\mu <a$ and unstable for $\mu >a$. The discriminant $(\mu -a{)}^2-8a$ distinguishes a node from a focus; its zero does not destroy hyperbolicity and is not another local bifurcation curve.

The local bifurcation curves areAt $(0,0)$ the linearization has a double zero eigenvalue and is nilpotent, giving a Bogdanov–Takens bifurcation. For $\mu \ne 0$ on $\lambda =0$, solving the local center manifold gives $\dot{x}=(\lambda -x^2)/\mu +\cdots$, displaying the saddle-node bifurcation. For $\mu <0$ the nonsaddle branch born at positive $\lambda$ is attracting; for $\mu >0$ it is repelling. On the Hopf curve the angular frequency is $\sqrt{2a}$. The permitted subcritical Hopf bifurcation creates a repelling periodic orbit on the stable-focus side $\mu <a$.

A local analysis alone does not say how far that periodic orbit extends. The global calculation below supplies the additional division of the stable-focus region. The complete nearby phase portraits and the leading global curve are shown together here. The $lambda<0$ flow has no closed periodic orbit, since a closed orbit would have index one and hence enclose an equilibrium point.

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.

At the saddle equilibrium $x=a=\sqrt{\lambda }$, put $T=\mu +a$. The two eigenvalues and saddle index areOn the nearby saddle-loop curve, $T=12a/7+O(a^2)>0$. Therefore $0<\delta <1$, with the more explicit expansionThe positive saddle quantity $m_{-}+m_{+}=T$ makes the loop repelling. In a transverse section its leading return map has the form $s\mapsto C{s}^{\delta }$; its derivative grows without bound as $s\downarrow 0$. This is the mechanism by which a positive saddle quantity makes a nearby saddle loop repelling. It agrees with the repelling periodic orbit coming from the subcritical Hopf bifurcation, rather than predicting an attracting cycle at the collision.

Write $F={\kappa }_{\mathrm{fold}}(\lambda )$, $G={\kappa }_{\mathrm{hom}}(\lambda )$ and $H=-\lambda$ for $\lambda <0$, so $0<F<G<H$ near the double-zero point. The six open regions are:

The changes III–IV, IV–V and V–VI are respectively a fold of periodic orbits, the double homoclinic event, and the two subcritical Hopf bifurcations. Thus the unstable periodic orbits from the subcritical Hopf bifurcations, the expanding saddle equilibrium passage, and the attracting periodic orbit from the other Hopf bifurcation fit into one consistent portrait.

Set $A=s-1\ne 0$ and rescale $q=\sqrt{|A|}u$, $p=\dot{q}$. The reflection-symmetric cubic double-zero normal form becomesIt is invariant under $(q,p)\mapsto (-q,-p)$. Thus the large-$s$ case has $\sigma =1$ and the small-$s$ case $\sigma =-1$. At the origin the Jacobian matrix has determinant $\lambda$ and trace $\kappa$: it is a saddle for $\lambda <0$, stable for $\lambda >0,\kappa <0$, and unstable for $\lambda >0,\kappa >0$. In the latter two cases the discriminant ${\kappa }^2-4\lambda$ distinguishes nodes from foci; its zero is not an equilibrium-creation bifurcation.

The nonzero equilibria satisfy $p=0$, $q^2=\lambda /\sigma$. At either of them the determinant is $-2\lambda$ and the trace is $\kappa -2\lambda$. Therefore for $s\gg 1$, a pair of saddles exists when $\lambda >0$, while for $s\ll 1$, a pair of attracting equilibria exists when $\lambda <0,\kappa <2\lambda$, and a repelling pair when $\lambda <0,\kappa >2\lambda$. The repelling or attracting points may be nodes or foci. Their origin collision occurs on the pitchfork bifurcation line $\lambda =0$, away from $\kappa =0$. For fixed $\kappa \ne 0$, the scalar centre equation there beginsFor $\kappa <0$, the large-$s$ case is a subcritical pitchfork and the small-$s$ case a supercritical pitchfork. For $\kappa >0$, the scalar criticality reverses, but there is already an unstable transverse eigenvalue, so the full-plane branches have the stability just listed.

At the origin, $\kappa =0$, $\lambda >0$ is a Hopf bifurcation with frequency $\sqrt{\lambda }$. Averaging the leading weakly nonlinear oscillator gives the amplitude equationConsequentlyThese are respectively a supercritical Hopf bifurcation and a subcritical Hopf bifurcation, with $\lambda >0$ fixed and the cycle sufficiently small. The portraits change from a stable focus to an unstable focus surrounded by a stable cycle in the first case; in the second, a stable focus is surrounded by an unstable cycle before the focus becomes unstable and that cycle disappears.

The small-$s$ case has an additional pair of Hopf bifurcations at the nonzero equilibria:Their criticality includes quadratic terms and is not read off from the positive $q^{2}p$ term alone. Set $q=a+x$, $a^2=-\lambda$, and $\delta =\kappa -2\lambda$. ThenFor an oscillator $\ddot{x}=-{\omega }_H^{2}x+A_2{x}^2+B_{2}x\dot{x}+C_3{x}^3+D_3{x}^2\dot{x}$, the cubic radial coefficient is $D_3/8+A_2{B}_2/(8{\omega }_H^2)$. Here it is $2/8+(-3a)(4a)/(8\cdot 2a^2)=-1/2$. Thus both nonzero Hopf bifurcations are supercritical: for small $\delta >0$, each newly unstable well equilibrium is surrounded by a stable small cycle with $r^2\sim \delta$.

To constrain periodic orbits, use either the divergence or the energyFor $\sigma =1,\kappa \le 0$, the divergence has one strict sign except on a line, and the Bendixson-Dulac criterion excludes every nonconstant periodic orbit. For $\sigma =-1,\kappa \ge 0$, the same reasoning excludes every periodic orbit. When the divergence changes sign, this criterion makes no such exclusion; the Hopf cycles above are consistent with that restriction. In the large-$s$ case a periodic orbit also requires $\lambda >0$: if $\lambda \le 0$, a positive maximum of $q$ would have $\ddot{q}=-\lambda q+q^3>0$, or a negative minimum would have negative acceleration, both impossible. For $\lambda >0$ the same extremum argument confines a cycle to $|q|<\sqrt{\lambda }$, between the two saddle locations. The local bifurcation lines and representative portraits on both sides are summarized below; their global completion requires part (c).

The larger bifurcation value is $\mu =0$. For $\mu <0$ the origin is stable and an unstable periodic orbit of radiussurrounds it. That orbit collapses into the origin as $\mu \uparrow 0$, after which the origin is unstable. This is a subcritical Hopf bifurcation.

After a positive rescaling of radius, its local normal form isor equivalentlyThe full quintic radial Hopf equation also has the stable outer branchIn the $(\mu ,r)$ bifurcation diagram, the branch $r=0$ is stable for $\mu <0$ and unstable for $\mu \ge 0$. The unstable $r_{-}$ branch runs from $(0,0)$ backwards towhere it folds into the stable $r_{+}$ branch; the latter continues to larger $\mu$.

At a planar saddle equilibrium with $m_{-}<0<m_{+}$, positive saddle quantity $m_{-}+m_{+}>0$ is equivalent to saddle index $\delta <1$. The local passage followed by the regular return has leading transverse map $s\mapsto C{s}^{\delta }$, $C>0$. Its derivative diverges as $s\downarrow 0$, so a nearby periodic orbit created by the saddle-loop splitting is repelling. This local mechanism agrees with the growth of the unstable periodic orbit born at a subcritical Hopf bifurcation in the quadratic Bogdanov–Takens unfolding.