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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.

The Hopf normal form is a local normal form of a dynamical system on the critical two-dimensional centre manifold. After shifting the equilibrium point and changing coordinates, its cubic truncation can be writtenwhere $\mu$ is the critical eigenvalue real part, $a=\mathrm{Re}\ell (0)$, $b=\mathrm{Im}\ell (0)$, and $\omega (0)>0$. Smooth parameter dependence of the cubic coefficients and higher-order terms are omitted in this truncation. The real coefficient $a$ controls radial saturation; the imaginary coefficient $b$ changes the angular frequency with amplitude. This finite-order reduction, rather than literal equality to a cubic equation everywhere, is the meaning of the Hopf normal form for a general system.^[ref]

In the cubic truncation, $r_{\ast }^2=-\mu /a$ whenever $\mu a<0$. Substituting into the angular equation gives $\Omega =\omega (\mu )-b\mu /a$. For the full smooth system, the leading normal form radius satisfies $r_{\ast }^2=-\mu /a+o(|\mu |)$ and the period tends to $2\pi /\omega (0)$. The exact circle and exactly constant angular frequency of the special model $a=-1$, $b=0$, $\omega (\mu )=\omega$ are conveniences of that model. In the original coordinates the small periodic orbit need not be a circle. Radial scaling can normalize a nonzero $a$ to $-1$ or $+1$, so $\sqrt{|\mu |}$ is a normalized leading amplitude, not a universal physical radius.

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.

The third planform is fixed by a diagonal reflection and changes sign under a quarter-turn. Extend the preceding group representation byThe other diagonal reflection changes only the sign of $B$. Consequently the first two components of an equivariant dynamical system have factors $A$ and $B$, whereas its third component is even in each of $A,B$. The quarter-turn exchanges $A^2$ and $B^2$ while negating $C$. Its admissible quadratic coupling is therefore $A^2-B^2$ in the third equation and opposite $AC,BC$ couplings in the first two. At cubic order the only new terms are $A{C}^2,B{C}^2,C^3,C(A^2+B^2)$. The linear action on the two-dimensional irreducible mode has one coefficient ${\mu }_1$, while the distinct scalar representation has its own coefficient ${\mu }_2$. A small splitting of the two marginal linear modes supplies these two unfolding parameters. With real coefficients the resulting amplitude equations areThese equations follow by listing all equivariant monomials through degree three; the displayed minus signs are coefficient conventions, not consequences of symmetry.

In the specified reduced system put $A=B=s$, $C=0$. A nonzero branch exists for ${\mu }_1>0$, with $q=s^2={\mu }_1/(1+\lambda )$. Write perturbations as $u=(\delta A+\delta B)/2$, $v=(\delta A-\delta B)/2$, and $c=\delta C$. Linearization givesFor the square-pattern interaction with a sign-changing scalar mode, the remaining eigenvalues have sum $\tau =p+{\mu }_2$ and product $D=p{\mu }_2+4{\alpha }_1{\gamma }_{1}q=p({\mu }_2+K)$, where $K=2{\alpha }_1{\gamma }_1/(\lambda -1)>0$. A real two-dimensional stability matrix has both eigenvalues in the left half-plane exactly when its trace is negative and its determinant is positive. ThereforeIf ${\mu }_2<-K$, the block has negative determinant and one growing direction, explaining the instability at sufficiently small ${\mu }_2$. At the lower endpoint ${\mu }_2=-K$, one eigenvalue is zero and the other is $p-K<0$. The symmetry $(A,B,C)\mapsto (B,A,-C)$ fixes the diagonal state and negates its critical $(v,c)$ direction. Thus a generic nonlinear unfolding has a pitchfork bifurcation to unequal-amplitude states with nonzero $C$.

At the upper endpoint ${\mu }_2=-p$, the trace vanishes andThe symmetric eigenvalue remains negative. The critical conjugate pair crosses the imaginary axis with derivative $d\mathrm{Re}\sigma /d{\mu }_2=1/2$. If its cubic Hopf normal form coefficient is nonzero, this is a Hopf bifurcation and nearby periodic solutions exist on one side. Their stability and the side of existence depend on that coefficient; the linear calculation does not determine them. It does establish the generic possibility of oscillating square patterns. The equality $p=K$ is a double-zero degeneracy, outside this nondegenerate Hopf bifurcation argument.

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

Put $x=1+y$ and retain linear terms. The delay differential equation is $\dot{y}(t)=-\alpha y(t-T)$, with characteristic equation of a delay differential equation $\lambda +\alpha e^{-\lambda T}=0$. Write $z=\lambda T$ and $q=\alpha T$. If a root has $\mathrm{Re}z\ge 0$, then $|z|=q{e}^{-\mathrm{Re}z}\le q$. Writing $z=a+ib$ gives $a=-q{e}^{-a}\cos b$. For $q<\pi /2$, $|b|<\pi /2$ and the right side is negative, contradicting $a\ge 0$. All characteristic roots therefore lie in the left half-plane.

An imaginary root $z=ib$ requires $\cos b=0$ and $b=q\sin b$. The first crossing is $q=\pi /2$, $z=\pm i\pi /2$. Implicit differentiation gives $dz/dq=-e^{-z}/(1+z)$, whose real part at the positive crossing is $(\pi /2)/(1+{\pi }^2/4)>0$. The later imaginary crossings occur at $q=\pi /2+2\pi k$ and also enter the right half-plane. Thus the equilibrium is locally exponentially asymptotically stable whenand unstable when $\alpha T>\pi /2$. At equality the linearized equation has neutral oscillatory modes; it is the Hopf bifurcation threshold. If stability is meant for the nonlinear equation itself, the borderline can also be resolved. In dimensionless time $s=t/T$, let $q_0=\pi /2$. A small center oscillation has the form $y=R\cos \theta +R^2(\cos 2\theta /5+\sin 2\theta /10)+O(R^3)$. Substitution at quadratic order gives these second-harmonic coefficients. At cubic order the fundamental forcing is $q_0{R}^3(\cos \theta /20+3\sin \theta /20)$. Projection onto the center mode divides the complex forcing by the characteristic derivative $1+i{q}_0$, yielding the Hopf normal form amplitude equationIts cubic coefficient is negative, so the equilibrium at equality is locally asymptotically stable, with algebraic rather than exponential decay. Thus the usual strictly decaying linear condition is $\alpha T<\pi /2$, while nonlinear local asymptotic stability includes $\alpha T=\pi /2$. The purely imaginary linear modes alone would not justify that borderline conclusion.

At a fixed point, the first equation gives $y=0$, the second then gives $x=0$, and the third gives $z=-\mu$. Thus there is exactly one equilibrium for every real $\mu$:The Jacobian matrix there isOne eigenvalue is $-1$. The other two have product one and sum $5ϵ\mu$. For $\mu <0$ their real parts are negative, so the equilibrium is asymptotically stable; for $\mu >0$ both real parts are positive, leaving two unstable directions and one stable direction. At $\mu =0$ the eigenvalues are $-1,i,-i$, withTherefore the only local bifurcation of this equilibrium is a Hopf bifurcation at $\mu =0$. A repeated real eigenvalue away from zero does not itself cause a bifurcation: the equilibrium remains hyperbolic there.

The cubic centre manifold calculation below gives the Hopf normal form radial coefficient $-9ϵ/8<0$, so this is a supercritical Hopf bifurcation. Small attracting periodic orbits occur on the $\mu >0$ side. In particular, at $\mu =0$ the leading cubic radial drift is inward, although linear stability alone cannot decide that boundary case.

Adjoin $\dot{\mu }=0$ and put $w=z+\mu$. The extended centre manifold is a graph $w=H(x,y,\mu )$ tangent to $w=0$ at the origin. The equations becomeHere $ϵ>0$ is fixed. The centre-manifold invariance equation, obtained by the chain rule, isWrite $H=H_2+O({\rho }^3)$, where $\rho =|x|+|y|+|\mu |$ and $H_2$ is a homogeneous quadratic polynomial. To quadratic order the centre-manifold invariance equation isThe terms $\mu x,\mu y,{\mu }^2$ have zero coefficients: their equations are homogeneous, and the operator $1+y{\partial }_x-x{\partial }_y$ is invertible on these quadratic polynomials. For $H_2=a{x}^2+bxy+c{y}^2$, comparison of coefficients givesHence $a=3/5$, $b=-2/5$ and $c=2/5$, givingSubstitution into the $y$ equation givesThe constant in the big O notation may depend on fixed $ϵ$. Thus the equations printed in part (b) are the retained terms of the local Taylor series, rather than an exact identity for the full reduced vector field.

For the Hopf bifurcation classification in part (a), set $\mu =0$ and use $x=r\cos \theta$, $y=-r\sin \theta$. The radial derivative is $\dot{r}=(x\dot{x}+y\dot{y})/r$. Its cubic angular mean isHere $⟨\cdot ⟩$ denotes the mean over one revolution. Removing the oscillatory cubic terms by a near-identity transformation gives this same cubic radial coefficient in the Hopf normal form. The linear radial coefficient is $5ϵ\mu /2$.

In a subcritical Hopf bifurcation, the small periodic orbit repels within the critical centre manifold and lies on the side where the equilibrium point attracts within that manifold. With the critical eigenvalue real part increasing through zero, the positive cubic sign gives the exact radial Hopf normal formFor $\mu <0$, the nonzero radius is $r_{\ast }=\sqrt{-\mu }$. Radii below $r_{\ast }$ shrink to zero, whereas radii above it grow. The radial derivative at the cycle is $\mu +3r_{\ast }^2=-2\mu >0$, proving that the periodic orbit repels. For example, $\mu =-0.04$ gives an unstable circle of radius $0.2$: a radius of $0.1$ decays while one of $0.3$ grows. The unstable circle shrinks into zero as $\mu \uparrow 0$; at $\mu =0$, $\dot{r}=r^3$ already drives every small positive radius outward.

For $\mu >0$ there is no nonzero constant radius in this cubic model, and zero repels. In a general dynamical system, this says only that the nearby orbits leave the local neighbourhood. It does not guarantee a distant attracting limit cycle, a jump to a particular amplitude, or hysteresis. Additional nonlinear or global information is needed; the quintic radial Hopf equation illustrates one possible stabilizing extension. Reversing the parameter orientation reverses which numerical side contains the branch, so the defining distinction between supercritical Hopf bifurcation and subcritical Hopf bifurcation is their reduced stability and their relation to the equilibrium point, rather than the bare sign of an arbitrarily named parameter.

In a supercritical Hopf bifurcation, a small periodic orbit attracts within the critical centre manifold on the side where the equilibrium point repels within that manifold. Orient the parameter so that its critical eigenvalue real part increases through zero. For the exact Hopf normal form $\dot{z}=(\mu +i\omega )z-|z{|}^{2}z$, with $\omega >0$, the nonzero constant radius solvesFor $0<r<r_{\ast }$ the radius grows; for $r>r_{\ast }$ it shrinks. A radial disturbance $\delta r$ therefore has leading equation $\dot{\delta r}=-2\mu \delta r$, since ${\partial }_r(\mu r-r^3){|}_{r_{\ast }}=-2\mu <0$. This is orbital stability: nearby orbits approach the circle, but their phase need not match a chosen representative. The exact angular frequency is $\omega$, so the period is $2\pi /\omega$ and the frequency is $\omega /(2\pi )$.

The Jacobian matrix at zero is $\left(\begin{array}{cc}\mu & -\omega \\ \omega & \mu \end{array}\right)$, with eigenvalues $\mu \pm i\omega$. Zero attracts for $\mu <0$ and repels for $\mu >0$. Exactly at $\mu =0$, $\dot{r}=-r^3$ givesThus zero still has asymptotic stability in this model, with algebraic decay, and there is no nonzero limit cycle at the threshold itself. As $\mu \downarrow 0$ on the cycle branch, both its radius $\sqrt{\mu }$ and its radial attraction rate $2\mu$ vanish. Small cycles consequently become slow to settle in radius even though their angular frequency stays fixed.

// Target: dynamical-systems.bigb