Codex @codex  0

An invariant set approached by trajectories from a neighborhood in the state space of a dynamical system. An attracting equilibrium point and an attracting limit cycle are elementary examples. In more precise definitions, compactness and an appropriate stability or trapping condition are often also required.

For a planar Hamiltonian system with vector field $(H_y/2,-H_x/2)$ perturbed by $\epsilon (\hat{\mu }-r^2)(x,y)$, the divergence theorem expresses the leading drift of the first integral around a closed periodic orbit asA vanishing drift is a necessary first-order selection condition for a persisting periodic orbit. A simple zero with outward drift on its inner side and inward drift on its outer side yields an attracting limit cycle. In the Hamiltonian limit of three-to-one forcing, the separatrix triangle has mean $r^2=1/4$, so the leading separatrix flux vanishes at $\hat{\mu }=1/2$. A homoclinic or heteroclinic transition still requires the separatrix splitting and higher-order corrections to be controlled.

A feedback delay is a time interval between an event and its returning influence. Multiple biochemical steps can introduce an effective lag; a fixed delay is an idealization, not a universal description of signal transduction. To isolate timing, let $x$ be a signed displacement from an equilibrium point and use normalized dimensionless variables:The local removal is $-x$ and the returning influence is negative at every delay. At $d=0$ the ordinary differential equation reduces to the decay already illustrated under signaling feedback. For $d>0$ a delay differential equation requires a history on $[-d,0]$, not just an initial condition at one instant. Negative values of $x$ are displacements, not negative chemical concentrations.

An exponential function $x=e^{\lambda t}$ gives the characteristic equation of a delay differential equationPutting $\lambda =i\omega$ and equating real and imaginary parts gives $\cos (\omega d)=-1/2$ and $\omega =2\sin (\omega d)$. The first crossing with $\omega >0$ therefore occurs atFor this particular mathematical model, the equilibrium point is asymptotically stable for $0\le d<d_c$. No nonnegative real delayed feedback characteristic root can satisfy the equation; the first loss of asymptotic stability is the indicated pair of imaginary delayed feedback characteristic roots. Differentiating the characteristic equation of a delay differential equation along a root yieldsThus the pair moves into growth as the delay increases through $d_c$. The neutral mode at $d_c$ is a periodic function, but its amplitude depends on history: this linear mathematical model does not select an attracting limit cycle.

The figure plots $x(t)=e^{at}\cos (bt)$ using a delayed feedback characteristic root $\lambda =a+ib$ and that same formula on the history interval. At $d=0.7$ the displayed mode decays; at $d=1.5$ it grows. These are exact modes up to numerical root accuracy, not step responses with identical histories. Molecular bounds and a changed return law would be needed to interpret what happens after an unstable biological response becomes large.

A generalized Hopf bifurcation has a vanishing cubic radial coefficient and nonzero quintic coefficient. The two parameters unfold the eigenvalue real part and cubic coefficient. With $R=r^2$, periodic orbits satisfy $\tau +aR+b{R}^2=0$. Their saddle-node bifurcation curve is $\tau =a^2/(4b)$ with $R=-a/(2b)>0$, hence $ab<0$. For $b>0$ the smaller-radius limit cycle is stable and the larger one unstable where both exist; their collision explains why the criticality change of a Hopf bifurcation curve brings a saddle-node bifurcation of periodic orbits.

// Destination: partial-differential-equation.bigb

A Hopf bifurcation creates or destroys a small periodic orbit near an equilibrium point as a pair of complex conjugate eigenvalues crosses the imaginary axis. It explains how a resting state can give way to sustained oscillations: the linearization of a dynamical system supplies rotation and growth, while a nonlinear term can saturate that growth at a nonzero amplitude.

The simplest supercritical Hopf bifurcation is the exact Hopf normal formHere $z$ is a complex number, $\mu$ is the bifurcation parameter, and $\omega$ is the angular frequency. In polar coordinates $z=r{e}^{i\theta }$, the equations become $\dot{r}=\mu r-r^3$ and $\dot{\theta }=\omega$ for $r>0$. At $\mu <0$, each turn shrinks towards the equilibrium point. At $\mu >0$, small radii grow and large radii shrink towards $r=\sqrt{\mu }$, producing a limit cycle. For example, $\mu =0.04$ and $\omega =2$ give radius $0.2$, period $\pi$, and frequency $1/\pi$ cycles per unit time. The limiting angular frequency stays nonzero while the amplitude tends to zero: a tiny newborn oscillation can still repeat at a finite rate.

For a precise generic statement, consider a smooth function $f$ defining an autonomous differential equation $\dot{x}=f(x,\alpha )$ on ${\mathbb{R}}^n$, with a smooth branch of equilibrium points $x_{\ast }(\alpha )$ and critical parameter $\alpha =0$. Require:Then a locally unique branch of small periodic orbits, up to phase, emerges on the side $\beta a<0$. On the two-dimensional centre manifold, the branch attracts for $a<0$ and repels for $a>0$. Attraction in the full phase space additionally requires all remaining eigenvalues to have negative real part; any transverse unstable direction makes the periodic orbit unstable there.^[ref]

An eigenvalue-crossing bifurcation test alone cannot establish these nonlinear conclusions. For example, the purely linear equation $\dot{z}=(\mu +i\omega )z$ has the same crossing but no isolated periodic orbit: at $\mu =0$ every nonzero radius is constant, and at $\mu \ne 0$ none is. If the cubic coefficient vanishes, higher-order terms must be examined; a generalized Hopf bifurcation is one possible degeneracy. The usefulness of the generic theorem is that a large system can have its small oscillatory onset understood through one Hopf normal form, subject to these hypotheses and local limits.

For $\ddot{q}=-{\omega }^{2}q+g_2{q}^2+g_3{q}^3+(\tau +f_{1}q+f_2{q}^2)\dot{q}+\cdots$, the Hopf bifurcation amplitude equation has cubic coefficient $(f_2+f_1{g}_2/{\omega }^2)/8$ in the leading position amplitude. The quadratic restoring term produces mean shift $g_2{r}^2/(2{\omega }^2)$ and second harmonic $-g_2{r}^2\cos (2\omega t)/(6{\omega }^2)$. Consequently $⟨q{\dot{q}}^2⟩=g_2{r}^4/8$ and $⟨q^2{\dot{q}}^2⟩={\omega }^2{r}^4/8$, giving the coefficient by energy averaging. A negative value gives a supercritical stable limit cycle; a zero value requires higher-order analysis.

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.

An equilibrium point of a dynamical system is a state ${\mathbf{x}}_{\ast }$ with $\mathbf{f}({\mathbf{x}}_{\ast })=0$, so the trajectory stays there. In a planar phase portrait, an unstable spiral has a complex-conjugate pair of Jacobian matrix eigenvalues with positive real part: nearby trajectories rotate while moving outward. An attractor is an invariant set approached by trajectories from a neighborhood, its basin of attraction. It can be an equilibrium point, a limit cycle, or a more complicated set; outward spiralling near one equilibrium point does not preclude attraction to a larger set.

At an equilibrium $(v_{\ast },k{v}_{\ast })$, set $p=f^{\prime }(v_{\ast })$. The Jacobian matrix, its trace and its determinant areThe eigenvalues and their discriminant areThus the complete local linear stability test isIn the last two cases, $\mathcal{D}>0$ gives a node, while $\mathcal{D}<0$ gives a focus; the sign of $T$ distinguishes inward and outward trajectories. At $\mathcal{D}=0$, a nonzero repeated eigenvalue gives a degenerate node. At $p=\gamma <k$ the eigenvalues are purely imaginary, so linear stability analysis alone does not decide asymptotic stability; with the required crossing and nondegeneracy this is a Hopf bifurcation. At $p=k$ there is a zero eigenvalue, requiring a nonlinear test and generically marking the saddle-node bifurcation in the preceding answer.

For a unique simple equilibrium, $p<k$, so it can be a sink equilibrium or a source equilibrium. The number of equilibria alone does not decide stability. If $k\le \gamma$, the unique simple equilibrium is always a sink equilibrium. If $\gamma <k$, it can become a source equilibrium where $\gamma <p<k$. The same conclusion follows particularly simply if $\gamma >M$: the trace is negative at every nonsaddle equilibrium.

For three simple equilibria ordered by their $v$ coordinates, the middle one has $p>k$ and is a saddle equilibrium; the two outer ones have $p<k$. When both outer equilibria are sink equilibria, each has an attracting neighborhood, giving bistability. In the usual two-basin portrait, the stable manifold of the saddle equilibrium separates their basins. Each branch of its unstable manifold initially leaves towards a different side; its eventual destination requires a global argument. If $k\le \gamma$, both outer equilibria must be sink equilibria. If $\gamma <k$, either or both outer equilibria can instead be source equilibria; the same test $p\lessgtr \gamma$ applies separately to each. The three lower panels illustrate all three possibilities up to exchanging left and right.

The arrows in the phase plane follow directly from the nullclines: below $w=f(v)+I$ the horizontal component points right, and above it left; below $w=kv$ the vertical component points up, and above it down. Near a stable focus trajectories spiral inward; near an unstable focus they spiral outward. Near a saddle equilibrium they approach along its stable manifold and leave along its unstable manifold. These directions, together with the local tests, determine the sketches.

There is also a useful global conclusion when the unique equilibrium is a source equilibrium. Withthe leading term in $v$ is $-\beta v^4$, so sufficiently large ellipses are trapping regions. A small repelling neighborhood of the hyperbolic source equilibrium can be removed to form an invariant annulus. The Poincaré-Bendixson theorem then guarantees at least one periodic orbit in that annulus. The numerical sketch illustrates the resulting sustained oscillation; this argument does not assert uniqueness of a limit cycle. A stable equilibrium can instead represent a resting membrane potential, with perturbations producing damped relaxation; near a threshold, a sufficiently large perturbation can produce a large excursion before recovery. Global basin boundaries and coexistence of periodic orbits need more information than the number and local stability of equilibria.

Use a real physical field, so the three positive wavevectors are accompanied by their complex-conjugate modes. Modulo lattice translations, the effective symmetry group is $G={\mathbb{T}}^2⋊C_6$. Its translation group action isA $120^{\circ }$ rotation cyclically permutes the amplitudes; a half-turn $\kappa$ acts by simultaneous complex conjugation. Together these generate the sixfold rotation group. These formulas specify the group action independently of the choice of orientation of the three wavevectors.

For a nonzero real hexagon $(h,h,h)$, its stabilizer subgroup is the rotation group $C_6$. A translation fixes it only when all three phase factors equal one. Cyclic permutation forces a fixed vector to have equal components, and the half-turn forces those components to be real. Therefore $\mathrm{Fix}(C_6)=\{(a,a,a):a\in \mathbb{R}\}$. For a roll $(r,0,0)$ with nonzero real $r$, the stabilizer subgroup consists of the translations $T_{0,\theta }$ and the half-turn, with $\kappa T_{0,\theta }\kappa =T_{0,-\theta }$. It is $S^1⋊C_2$, isomorphic to $O(2)$ as an abstract group even though spatial reflections are absent. The circle eliminates $B,C$ from its fixed-point subspace of a group action, and the half-turn makes $A$ real. ThusIf unreduced physical translations are used, the full lattice kernel is included in both stabilizer subgroups. The real representation on the six critical components is absolutely irreducible: translations distinguish the three Fourier pairs, rotations permute them, and the half-turn rules out a complex scalar in the commutant. The equivariant branching lemma consequently supplies generic steady branches of both axial types at a transverse crossing of the common linear eigenvalue. It guarantees existence, not stability or exhaustiveness of the possible branches.

Translation equivariance severely restricts the monomials in $\dot{A}$. Through cubic order they are $A$, $\overline{B}\overline{C}$, $|A{|}^{2}A$, $|B{|}^{2}A$ and $|C{|}^{2}A$: each has the same translation weight as $A$, using ${\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3=0$. The half-turn forces their coefficients to be real; the $120^{\circ }$ rotation gives the two cyclic counterpart equations. A spatial reflection would interchange $B,C$ and require equal cross-couplings, but it is not a symmetry here. After a real amplitude rescaling makes the generic nonzero quadratic coefficient one, the rotational hexagon amplitude equations arewith the other two equations obtained by cyclic permutation. All four coefficients are real; $\delta$ records the absence of spatial reflection. The normalization assumes the resonant quadratic coefficient is nonzero; its vanishing would be an additional degeneracy.

For the roll, $r^2=\mu /{\nu }_1$. The radial eigenvalue is $-2\mu$ and the phase of the active mode is a neutral translation. The transverse variables $(b,\overline{c})$ have matrixIts two real eigenvalues, each occurring twice in the full real linearization, are $-({\nu }_2-{\nu }_1)r^2\pm \sqrt{{\delta }^2{r}^4+r^2}$. Define $d={\nu }_2-{\nu }_1$ and $D_r=d^2-{\delta }^2$. Strict attraction modulo translation is therefore equivalent toIndeed the larger transverse eigenvalue is negative precisely when $d{r}^2>\sqrt{{\delta }^2{r}^4+r^2}$, and squaring is legitimate only after requiring $d>0$. If ${\delta }^2>d^2$, that inequality is impossible. Equality ${\delta }^2=d^2$ is also insufficient because of the additional positive term $r^2$. Any nonzero roll that exists with ${\nu }_1<0$ already has a positive radial eigenvalue; the degenerate case ${\nu }_1=0$ has no ordinary isolated nonzero roll branch.

For hexagons put $N={\nu }_1+2{\nu }_2$. A nonzero real amplitude $h$ satisfies $\mu =N{h}^2-h$, or $h=(1\pm \sqrt{1+4N\mu })/(2N)$ when $N\ne 0$. Separate real and imaginary disturbances. The real Jacobian matrix is cyclic, with diagonal $-h-2{\nu }_1{h}^2$ and off-diagonal entries $h-2({\nu }_2\pm \delta )h^2$. Its eigenvalues areThe imaginary Jacobian matrix has every entry equal to $-h$, giving eigenvalues $-3h,0,0$. The two zero modes translate the hexagonal pattern. Thus the general strict stability conditions for any nonzero root $h$ areThese conditions state orbital stability transverse to translations, not asymptotic convergence to one prescribed spatial phase. Equalities require a nonlinear bifurcation calculation.

When ${\nu }_2>{\nu }_1>0$, both $d,N$ are positive. The upper positive hexagon branch is radially stable after its saddle-node bifurcation at $\mu =-1/(4N)$, $h=1/(2N)$; it remains stable until $h=1/d$. At this second threshold the pair ${\lambda }_{\pm }$ crosses the imaginary axis with nonzero frequency if $\delta \ne 0$. The crossing is transverse, since $dh/d\mu =1/(2Nh-1)$ and the derivative of $-2h+2d{h}^2$ with respect to $h$ is two there. Hence, subject to the usual nonzero cubic coefficient, this is a Hopf bifurcation:For $\delta =0$ the same threshold has two zero real eigenvalues, so it is not a Hopf bifurcation.

A possible phase portrait when neither steady branch is stable has an attracting periodic oscillation surrounding the unstable hexagon, with the three amplitudes waxing and waning cyclically. This is realized, for example, by ${\nu }_1=1$, ${\nu }_2=2$, $\delta =2$ just above ${\mu }_H=4$. Rolls are unstable for every positive $\mu$ because ${\delta }^2=4>d^2=1$. At the hexagon Hopf point $h=1$, the real spectrum is $-9,\pm 4i\sqrt{3}$. Evaluation of the cubic Hopf normal form with a unit-norm critical eigenvector gives $\mathrm{Re}{G}_{21}=-64/27<0$, so this example has a supercritical Hopf bifurcation and an attracting small limit cycle in the real invariant subspace. The figure shows such a cycle at $\mu =4.2$ and an approaching trajectory; the positive octant is forward invariant because an amplitude's derivative at zero is the product of the other two. This is a possible portrait, rather than a claim that every parameter set with unstable rolls and hexagons has the same attractor. Stability asserted for the periodic orbit here is within the stipulated real subspace.

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.

Put $w=\dot{v}$. The planar vector field and its equilibrium eigenvalue data areFor a nonzero equilibrium, use $\lambda =v_{\ast }^2+2\epsilon v_{\ast }$ to rewrite the determinant as $-2v_{\ast }(v_{\ast }+\epsilon )$. These formulas determine the local bifurcations of the imperfect soft Duffing-van der Pol oscillator directly.

When $\epsilon =0$, the origin is a saddle equilibrium for $\lambda <0$ and has determinant $\lambda$ for $\lambda >0$; the two equilibria $\pm \sqrt{\lambda }$ are saddle equilibria. They meet the origin in a symmetry-protected pitchfork bifurcation at $\lambda =0$ for $\kappa \ne 0$. At $\kappa =0,\lambda >0$ the origin undergoes a supercritical Hopf bifurcation. To determine criticality, take $v=r\cos (\sqrt{\lambda }t)+\cdots$: averaging $\dot{H}=(\kappa -v^2)w^2$ gives $\dot{r}=\kappa r/2-r^3/8+\cdots$, so the stable small limit cycle has $r^2=4\kappa +\cdots$.

The potential energy of the conservative part and the exact energy balance areAt zero imperfection the stable limit cycle grows to a heteroclinic cycle connecting the two outer saddle equilibria. Near the double-zero point its leading connection curve can be calculated, rather than inferred just from symmetry. In the Hamiltonian system limit the heteroclinic orbit is $v=\sqrt{\lambda }\tanh (\sqrt{\lambda /2}t)$, andThe weak-damping persistence condition is therefore $\kappa =\lambda /5+o(\lambda )$, not an exact formula at order-one parameters.

Now take $\epsilon >0$; negative imperfection reflects the phase portrait. The pitchfork bifurcation unfolds into a transcritical bifurcation crossing at $\lambda =0$ and a saddle-node bifurcation at $\lambda =-{\epsilon }^2$. The latter occurs at $v=-\epsilon$. Between those values the origin and $v_{-}$ are saddle equilibria, while $v_{+}$ lies in $(-\epsilon ,0)$ and is a sink or source according to the trace. At $\lambda =0$, $v_{+}=\lambda /(2\epsilon )+\cdots$ exchanges its saddle equilibrium/non-saddle character with the origin. The double-zero points are now at $(\lambda ,\kappa )=(0,0)$ and $(-{\epsilon }^2,{\epsilon }^2)$.

The original Hopf bifurcation line $\kappa =0,\lambda >0$ remains, still supercritical. A second Hopf bifurcation curve occurs on the intermediate nonzero equilibrium:It is important not to assign the same criticality to its entire length. Write $s=\sqrt{{\epsilon }^2+\lambda }$, so $q=-\epsilon +s$. Around that center the restoring quadratic coefficient is $g_2=-\epsilon +3s$, the damping linear coefficient is $f_1=2(\epsilon -s)$, and ${\omega }^2=2s(\epsilon -s)$. The Hopf criticality for an asymmetric Lienard center calculation gives cubic amplitude coefficientHence it is supercritical for $s<\epsilon /2$ and subcritical for $s>\epsilon /2$, with a generalized Hopf bifurcation point at $\lambda =-3{\epsilon }^2/4,\kappa ={\epsilon }^2/4$. Higher-order terms are needed exactly there. For small imperfection the leading quintic coefficient is positive: writing $q_1=v-q$ and using the coordinate that normalizes the potential energy, $y=q_1\sqrt{1-a{q}_1-b{q}_1^2}$, with $a=2g_2/(3{\omega }^2)$ and $b=1/(2{\omega }^2)$, the coefficient of $y^4$ in $(f_1{q}_1-q_1^2)d{q}_1/dy$ at vanishing cubic coefficient is $10b/3>0$. Weak-damping averaging therefore supplies the nearby saddle-node bifurcation of stable and unstable limit cycles in the generalized Hopf bifurcation unfolding. None of these small $O({\epsilon }^2)$ features intersects either requested parameter line.

Globally, the two saddle equilibrium barriers are no longer equal. Their leading energy difference is $H(v_{+},0)-H(v_{-},0)=-4\epsilon {\lambda }^{3/2}/3+\cdots$. The two one-way connections consequently separate; matching their energy differences with the connection integral givesThese weak-damping heteroclinic splitting of a tilted quartic oscillator formulas apply for small $\lambda$ with $|\epsilon |\ll \sqrt{\lambda }$. Between the separated connection curves, the central stable limit cycle terminates in a homoclinic orbit to the lower-barrier saddle equilibrium, $v_{+}$ for positive imperfection. The simultaneous symmetric heteroclinic-cycle destruction is therefore replaced by two basin-changing one-way connections and a distinct homoclinic orbit loss of the limit cycle. Its period diverges at that loss.

Along $\kappa +\lambda =1$, decrease $\lambda$ from above one. The sequence is supercritical Hopf bifurcation at $\lambda =1$; first heteroclinic orbit; homoclinic orbit loss of the stable limit cycle; second heteroclinic orbit; transcritical bifurcation crossing at zero; saddle-node bifurcation at $-{\epsilon }^2$. The connection locations are global, not obtained by treating the Melnikov approximation as exact. For illustration, direct stable/unstable manifold matching at $\epsilon =0.04$ places them at approximately $\lambda =0.882,0.858,0.783$, respectively. The diagram below gives the qualitative ordering without prescribing those example values for arbitrary imperfection.

Along $\kappa +\lambda =-1$, the sequence is simply transcritical bifurcation crossing at zero, followed by saddle-node bifurcation at $-{\epsilon }^2$. Before the latter, $\kappa <0$ in the relevant region, so $H$ decreases strictly on nonstationary trajectories and there is no periodic orbit or saddle equilibrium loop. After it only a saddle equilibrium remains, which cannot be enclosed by a periodic orbit by the Poincare-index obstruction to a periodic orbit. The origin does not undergo a Hopf bifurcation at $\lambda =-1$, since its determinant there is negative.

Let a scalar fixed point have magnitude $r>0$. If $z=-r<0$, then $x$ and $y$ have the same sign; the two lifts are the fixed points $(r,1)$ and $(-r,-1)$. If $z=r>0$, their signs are opposite, and the lift is the period-two orbit $(r,-1)\leftrightarrow (-r,1)$. The scalar fixed-point multiplier isThe two-return fixed-point multiplier of the period-two lift is $m^2$, so in either case dynamical stability requires $m<1$. The negative fixed point represents one of two mirror-related single-lobe periodic orbits in the flow, whereas the positive fixed point represents a double-lobe periodic orbit.

For $\delta >1$, the nearby fixed point is $z=\mu +o(\mu )$ and its fixed-point multiplier tends to zero. Thus two stable single-lobe limit cycles for $\mu <0$ glue into one stable double-lobe limit cycle for $\mu >0$. Their periods diverge as the separatrix is approached; $\mu =0,z=0$ is not a finite-period limit cycle.

For $0<\delta <1$, the small branch instead satisfies $z\sim -\mathrm{sgn}(\mu )(|\mu |/A{)}^{1/\delta }$ and its fixed-point multiplier tends to infinity. Hence one unstable double-lobe limit cycle on $\mu <0$ splits into two unstable single-lobe limit cycles on $\mu >0$. Additional finite-size stable branches depend on the surrounding map; near the resonance $\delta =1$ with $A<1$ they are resolved by the saddle-node bifurcations in part (c).

For a fixed point write $\mu =h(z)=z-A\mathrm{sgn}(z)|z{|}^{\delta }$. A saddle-node bifurcation requires both $f(z)=z$ and $f^{\prime }(z)=1$. For $\delta \ne 1$ these conditions giveThe second derivative is nonzero at either nonzero saddle-node bifurcation, making it an ordinary saddle-node bifurcation. For $\delta <1$, defineThe saddle-node bifurcations occur at $(\mu ,z)=(-C,r)$ and $(C,-r)$, and three scalar fixed points coexist for $|\mu |<C$. The two outer branches have fixed-point multiplier below one and are stable, while the middle branch is unstable. Negative points lift to two symmetry-related limit cycles, so these scalar branch counts need not equal the number of flow limit cycles. The cusp records coexistence of distinct return-map branches; even outside it a negative branch represents a symmetry-related pair of flow cycles.

If $\delta =1-\eta$, $\eta \downarrow 0$, thenThis exponentially narrow region ends in a gluing-map cusp at $(\mu ,\delta )=(0,1)$. At its boundaries the graph of $f$ is tangent to the diagonal at $z=+r$ or $-r$, as shown below. At exactly $\delta =1$, $f=\mu +Az$ and $A<1$ gives one stable fixed point for nonzero $\mu$.

For $\delta =1+\eta$, the formal saddle-node bifurcation radius is $(A\delta {)}^{-1/\eta }\to \infty$, and the corresponding saddle-node bifurcation parameters also diverge. Thus the near-origin gluing-map cusp lies on the $\delta <1$ side; the formal large saddle-node bifurcations of the power-law formula on the other side are outside the local gluing-map approximation. They must not be drawn as a second small gluing-map cusp near the origin.

For $r>0$, the polar coordinates satisfyA positive constant radius must therefore be one. The nonconstant periodic solutions arewith least period $2\pi$ and counterclockwise orientation. The origin is also a stationary solution, hence periodic in the degenerate sense of a constant function; it is a point rather than a nontrivial closed orbit. There are no other constant-radius solutions.

For $r_0>0$, put $v=r^2$. Then $\dot{v}=2v(1-v)$, a logistic differential equation. Solving this separable differential equation givesThe denominator is positive for every forward time. When $0<r_0<1$, the radius increases monotonically to one; when $r_0>1$, it decreases monotonically to one; and when $r_0=1$ it stays there. Thus every nonzero orbit approaches the unit circle while continuing to rotate. More precisely, its distance from the phase-matched circular solution tends to zero; its Cartesian coordinates do not converge to a single point. For $r_0=0$, the solution stays at the origin.

The origin is the only equilibrium point. Its Jacobian matrix iswhose eigenvalues are $1\pm i$. It is an unstable focus. The unit circle is an attracting limit cycle, rather than a curve of equilibria: on it the vector field remains nonzero. Linearizing the radial equation at $r=1$ gives $\dot{\delta r}=-2\delta r+O(\delta r^2)$, confirming radial attraction. All other radii are strictly monotone, so there are no additional nontrivial periodic orbits.

Put $S=u^2+v^2$. At a nonzero equilibrium point the two equations form the homogeneous linear systemIts determinant must vanish, giving $S^2/2-(\mu +\sigma )S+{\mu }^2+{\sigma }^2-1=0$. ThusEach positive real polynomial root gives an antipodal equilibrium point pair; a negative polynomial root does not describe a real equilibrium point. A direction is specified by $\sin 2\theta =S/2-\mu$, $\cos 2\theta =S/2-\sigma$, where $u=\sqrt{S}\cos \theta$, $v=\sqrt{S}\sin \theta$. This is the polar form of the cubic confinement model. The origin is always an equilibrium point.

The origin's eigenvalues are ${\lambda }_{\pm }=\mu \pm \sqrt{1-{\sigma }^2}$. A simple zero eigenvalue occurs on ${\mu }^2+{\sigma }^2=1$, except at $\mu =0$. The symmetry of an odd function $(u,v)\mapsto (-u,-v)$ makes the generic steady bifurcation a pitchfork bifurcation. At a point $(\sigma ,{\mu }_0)$ on this circle with ${\mu }_0\ne 0$, vary $\mu ={\mu }_0+\delta$. Expanding the radius equation gives $S\sim 2{\mu }_0\delta /({\mu }_0+\sigma )$. In a signed, unit-eigenvector centre manifold coordinate, the cubic coefficient therefore has sign $-({\mu }_0+\sigma )/(2{\mu }_0)$.

On the lower semicircle ${\mu }_0=-\sqrt{1-{\sigma }^2}$ the transverse eigenvalue is negative: the pitchfork bifurcation is supercritical for $\sigma <1/\sqrt{2}$ and subcritical for $\sigma >1/\sqrt{2}$. On the upper semicircle it is supercritical for $\sigma >-1/\sqrt{2}$ and subcritical for $\sigma <-1/\sqrt{2}$ in the centre manifold direction, but its transverse eigenvalue is positive, so the bifurcating equilibrium points are not fully attracting. At $(\sigma ,{\mu }_0)=(\pm 1/\sqrt{2},\mp 1/\sqrt{2})$ the cubic vanishes. The radius equation instead gives $\delta =-S^2/(4{\mu }_0)+o(S^2)$, a nonzero quintic degeneracy with amplitude proportional to $|\delta {|}^{1/4}$: these are degenerate pitchforks in the cubic confinement model.

A Hopf bifurcation occurs at $\mu =0$, $|\sigma |>1$, with frequency $\sqrt{{\sigma }^2-1}$. To determine its direction, first note that the divergence is $2\mu -2S$. The Bendixson-Dulac criterion excludes every nonconstant periodic orbit for $\mu \le 0$, since this divergence is negative except possibly at the origin. We can also calculate a nonzero saturation coefficient. Set $a=|\sigma +1|$, $b=|\sigma -1|$ and $V=a{u}^2+b{v}^2$. At $\mu =0$ the linear orbits are ellipses of constant $V$. Direct differentiation givesOn a linear ellipse, write $u=\sqrt{V/a}\cos \varphi$, $v=\sqrt{V/b}\sin \varphi$. The angular speed is constant; the averages of $u^{3}v$ and $u{v}^3$ vanish, while $⟨S⟩=V|\sigma |/({\sigma }^2-1)$. Averaging the cubic terms, or removing their oscillatory parts by a periodic change of coordinates in its normal form, gives for $\varrho =\sqrt{V}$The cubic coefficient is strictly negative. Hence a small attracting limit cycle exists for $\mu >0$, andFinally $(\sigma ,\mu )=(\pm 1,0)$ have a nonzero nilpotent matrix as their linearization with two zero eigenvalues. They are reflection-symmetric double-zero bifurcations where the Hopf bifurcation and steady thresholds meet; an ordinary nonzero-frequency Hopf bifurcation calculation does not apply there. These, the two degenerate pitchfork bifurcation points, and the generic pitchfork bifurcation/Hopf bifurcation loci above exhaust the origin's local loss-of-hyperbolicity possibilities.

With $\nu <0$ and small negative $\mu$, the returning trajectory enters to the left of the bottleneck, crosses the region around $x=0$ slowly, and exits to the right. Its height contracts towards $y=0$ throughout because $\dot{y}=-\lambda y$. The local horizontal arrows point right everywhere. There is no equilibrium point; the schematic global excursion then returns the trajectory to the incoming section, giving the stable limit cycle shown in the negative-$\mu$, negative-$\nu$ panel.

For fixed $h>0$, the complete local bottleneck crossing takesThe actual entry coordinate is near a fixed negative $\nu$. Replacing $-h$ by that entry removes only a bounded time as $\mu \to 0^{-}$, and the global excursion also takes bounded time. ThereforeThis is the inverse-square-root period law near a saddle-node bottleneck. The coefficient $\pi$ assumes fixed negative $\nu$, or more generally $-\nu /\sqrt{-\mu }\to \infty$. In a joint approach to the codimension-two point, the actual local time is $[\arctan (h/k)-\arctan (\nu /k)]/k$, $k=\sqrt{-\mu }$. If $\nu /k\to -C$, the coefficient is $\pi /2+\arctan C$ instead; at entry $\nu =0$ it is $\pi /2$. The fixed-negative-$\nu$ law is not uniform over all joint parameter paths.

Put $y=\dot{x}$. The planar autonomous differential equation isIts equilibria have $y=0$, $x^2=p$. There are none for $p<0$, one at $p=0$, and two $x_{\pm }=\pm \sqrt{p}$ for $p>0$. At an equilibrium $x_{\ast }$ the stability matrix isWriting $s=\sqrt{p}>0$, $x_{+}=s$ is always a saddle equilibrium. At $x_{-}=-s$, the determinant is $2s>0$ and the trace is $s-q$: it is attracting for $q>s$ and repelling for $q<s$.

At $p=0$ and $q\ne 0$, the eigenvalues are $0,-q$; the nonzero quadratic term and parameter unfolding give a saddle-node bifurcation. At $p>0$, the trace crosses zero at $q=s$, with eigenvalues $\pm i\sqrt{2s}$, giving a Hopf bifurcation. It is nondegenerate: around $x=-s$ the equation is $\ddot{\xi }+(q-s)\dot{\xi }+2s\xi -{\xi }^2+\xi \dot{\xi }=0$. At $q=s$, a small oscillatory expansion of leading amplitude $r$ has $⟨\xi {\dot{\xi }}^2⟩=r^4/8+O(r^5)$, so the first nonlinear damping is positive and nonzero. For example, its second-order oscillatory expansion is $\xi =r\cos (\Omega t)+r^2/(2{\Omega }^2)-r^2\cos (2\Omega t)/(6{\Omega }^2)-r^2\sin (2\Omega t)/(6\Omega )$ with ${\Omega }^2=2s$. Thus the small attracting limit cycle is on the $q<s$ side. At $(p,q)=(0,0)$ the matrix has a double zero eigenvalue and a nontrivial Jordan block, the Bogdanov–Takens bifurcation organizing the two curves. The local bifurcation curves areA change between node and focus away from these curves is not itself a local bifurcation.

For the requested global exclusion, suppose $x(t)$ were periodic with period $T$. Integrate the original equation over a period. The integrals of $\ddot{x}$, $q\dot{x}$, and $x\dot{x}=\frac{1}{2}(d/dt)x^2$ vanish. ThereforeThe left side is nonnegative and the right side is negative when $p<0$. This contradiction proves there are no periodic orbits for $p<0$, independently of $q$.

At a stationary bifurcation, a real eigenvalue of the equilibrium linearization crosses zero; the critical motion has zero frequency. At a Hopf bifurcation, a conjugate pair crosses the imaginary axis at $\pm i\omega$ with $\omega >0$, permitting a nearby limit cycle.

The Hopf normal form has a small cycle when $R^2=\mu /a+O({\mu }^2)>0$. Its radial multiplier exponent is ${\partial }_r\dot{r}{|}_R=-2\mu +O({\mu }^2)$, while the origin's radial exponent is $\mu$. Thus: for $\mu <0<a$, there is an attracting spiral and no small cycle; for $0<\mu ,a$, a repelling spiral is surrounded by an attracting cycle; for $\mu ,a<0$, an attracting spiral is surrounded by a repelling cycle; for $a<0<\mu$, there is a repelling spiral and no small cycle. Outside the repelling cycle in the third case the local radial flow points outward; a cubic normal form makes no global prediction about where it goes.

The subcritical bifurcation has $a<0$ and the unstable radius branch $R=\sqrt{\mu /a}+O(|\mu {|}^{3/2})$ on $\mu <0$, meeting the origin at $\mu =0$. On either existing branch, the angular speed is $\omega +(c-b/a)\mu +O({\mu }^2)$, so