Bifurcation theory

Bifurcation theory studies qualitative changes in equilibria and invariant sets as parameters vary.

A subcritical branch emerges on the parameter side where the original state remains stable. In the cubic Landau amplitude equation $\dot{A}=\mu A-g|A{|}^{2}A$ with $g<0$, the small nonzero branch lies at $\mu <0$ and is unstable to radial amplitude perturbations. Higher-order saturation can generate a finite-amplitude stable branch and hysteresis, but this does not follow from a cubic truncation alone.

A supercritical branch emerges on the parameter side where the original state loses stability. In the cubic Landau amplitude equation $\dot{A}=\mu A-g|A{|}^{2}A$ with $g>0$, the nonzero branch $|A{|}^2=\mu /g$ lies at $\mu >0$ and is stable to radial amplitude perturbations. Continuous phase symmetry still leaves a neutral phase direction. The direction of a branch and its stability should both be checked in a general problem of bifurcation theory.

A codimension-two bifurcation requires two independent parameter conditions to hold simultaneously. Simultaneous stationary and Hopf bifurcation thresholds require both kinds of critical mode in the reduced amplitude equation. A crossing of modes at different wavenumbers is different from a zero-frequency double-zero degeneration of one fixed-wave-number cubic.

For a chosen orientation and phase-reduced steady amplitude $a$ and nonresonant oscillatory amplitude $z$, a schematic normal form is $\dot{a}=r_{s}a-g_s{a}^3-h_{s}a|z{|}^2$, $\dot{z}=(r_o+i{\omega }_0)z-g_o|z{|}^{2}z-h_o{a}^{2}z$. For $r_s,g_s>0$ the pure steady branch exists and is radially attracting; its oscillatory perturbation is damped when $r_o-h_{o,r}{r}_s/g_s<0$, where $h_{o,r}=\mathrm{Re}(h_o)$. For $r_o,g_{o,r}>0$, $g_{o,r}=\mathrm{Re}(g_o)$, the pure oscillatory branch exists and is radially attracting; its steady perturbation is damped when $r_s-h_s{r}_o/g_{o,r}<0$. The transverse inequalities are not existence or full stability tests by themselves. Mixed intensities $u=a^2>0$, $v=|z{|}^2>0$ solve $g_{s}u+h_{s}v=r_s$, $h_{o,r}u+g_{o,r}v=r_o$. Their intensity Jacobian matrix has trace $-2(g_{s}u+g_{o,r}v)$ and determinant $4uv(g_s{g}_{o,r}-h_s{h}_{o,r})$, giving attraction in intensities when the trace is negative and determinant positive. Resonant phases, oppositely travelling modes or zero-frequency limits need additional retained amplitudes; this schematic pair is not a universal complete normal form.

At a stationary bifurcation, a real eigenvalue of the linearized system passes through zero. Nearby equilibria may then be created, destroyed, or exchange stability.

An amplitude equation describes the slow evolution of the coefficient of a critical mode near an instability threshold. Symmetry and solvability determine its leading nonlinear terms.

A critical pattern Fourier mode transforms under a spatial translation as $A\mapsto e^{i\varphi }A$. A forcing at three times the critical wavenumber transforms as $F\mapsto e^{3i\varphi }F$. Their lowest resonant product with phase weight one is $F{\overline{A}}^2$. This determines the form of the leading weak-forcing amplitude equation, while projection onto an adjoint eigenfunction determines its coefficient. Symmetry permits this coupling but does not ensure that it is nonzero. Travelling forcing gives a time-dependent phase to $F$.

For small positive forcing-frame frequency $\omega$, use $\mu =\hat{\mu }{\omega }^2$, $C=\omega z$ and $\tau =\omega T$ in the three-to-one spatially forced amplitude equation. The leading system is $z_{\tau }+iz={\overline{z}}^2$. Writing $z=x+iy$ gives the Hamiltonian system $(x_{\tau },y_{\tau })=(H_y/2,-H_x/2)$ and the displayed first integral. The origin is a center equilibrium; three saddle equilibria at $(0,1)$ and $(\pm \sqrt{3}/2,-1/2)$ lie on $H=1/3$. The factorization $H-1/3=(1+2y)[x^2-(y-1{)}^2/3]$ reveals a triangular heteroclinic cycle, containing closed periodic orbits for every $0<H<1/3$.

The canonical three-to-one spatially forced amplitude equation has nonzero equilibrium points $C=R{e}^{i\theta }$ satisfying $\cos 3\theta =(R^2-\mu )/R$ and $\sin 3\theta =-\omega /R$. Thus $(R^2-\mu {)}^2+{\omega }^2=R^2$. Each positive amplitude has three phases separated by $2\pi /3$: two amplitude branches normally mean six complex equilibrium points. For $\mu >{\omega }^2-1/4$ the larger branch is asymptotically stable and the smaller consists of saddle equilibria, except that its zero-amplitude root at $(\mu ,\omega )=(0,0)$ is not a nonzero equilibrium point. Equality gives a saddle-node bifurcation. This phase locking breaks continuous translation symmetry down to threefold symmetry.

For the three-to-one spatially forced amplitude equation with forcing $e\exp [3i(\Omega T+\delta )]$, put $A=B\exp [i(\Omega T+\delta )]$. The real phase constant $\delta$ belongs inside the factor of $i$. If $c>0$ and the phase has been chosen so that $e>0$, the scaling $B=(e/c)C$, $\mathcal{T}=(e^2/c)T$ gives the displayed canonical amplitude equation, with $\mu =c\tilde{\mu }/e^2$ and $\omega =c\Omega /e^2$. A negative cubic saturation coefficient cannot give the same cubic sign under a forward-time normalization; zero forcing or zero cubic coefficient requires another scaling.

The real-coefficient Ginzburg–Landau amplitude equation has complex $A$ and real parameters, usually $\xi ,g>0$ near a supercritical stationary pattern onset. A detuned plane wave $A=\rho e^{iQX}$ has ${\rho }^2=(r-\xi Q^2)/g$. Its phase and amplitude sidebands determine the narrower Eckhaus instability stability band. It is a local envelope model; additional conserved fields or mean flows may need their own equations.

For $r,\xi ,g>0$, longitudinal phase modulation of a detuned convection roll yields diffusion coefficient $D_{\varphi }=\xi (r-3\xi Q^2)/(r-\xi Q^2)$. The expression follows by eliminating the fast amplitude correction from the coupled linear amplitude-phase equations at small modulation wavenumber. Thus convection rolls become unstable for $Q^2>r/(3\xi )$ although they exist for $Q^2<r/\xi$. At the boundary the leading phase diffusion vanishes and higher spatial orders matter.

The cubic Landau amplitude equationwith $g>0$ describes saturation through a supercritical pitchfork bifurcation when a reflection symmetry permits both signs of $A$.

A steady spatial forcing at twice the critical wavenumber permits coupling of the negative critical Fourier mode to the positive critical mode. A weakly nonlinear expansion then permits a term $\beta \overline{A}$ in the Landau amplitude equation. Its coefficient is found by projecting the resonant forcing-advection terms onto the adjoint eigenfunction. Reflection-symmetric forcing permits real coefficients. The symmetry permission does not prove a nonzero coefficient: it can vanish for a particular model or mode structure.

For real $\mu ,\beta$ and $\lambda >0$, writing $A=x+iy$ gives a gradient flow with potential $V=-(\mu +\beta )x^2/2-(\mu -\beta )y^2/2+\lambda (x^2+y^2{)}^2/4$. The origin has exponential asymptotic stability for $\mu <-|\beta |$, is algebraically attracting at equality, and is unstable above it. Nonzero stable equilibria are the real pair $\pm \sqrt{(\mu +\beta )/\lambda }$ for $\beta >0$, or the imaginary pair $\pm i\sqrt{(\mu -\beta )/\lambda }$ for $\beta <0$, when their squared amplitudes are positive. Differentiating the two real equations gives eigenvalues $-2(\mu +\beta ),-2\beta$ on the real branch and $-2(\mu -\beta ),2\beta$ on the imaginary branch. For $\beta =0$, $\mu >0$ gives a radially attracting circle with neutral phase, so individual equilibrium points have Lyapunov stability but are not individually asymptotically attracting.

In the free-slip Stokes flow temperature model, let the critical temperature be $g=\sin \pi z$ and vertical velocity $f=sg$, $s=k_c^2+{\pi }^2$. A forced positive second harmonic $(W,\Theta )e^{2i{k}_{c}x}$ couples to the negative critical harmonic. Its temperature solvability condition has integrandThe identity follows by substituting $f=sg$ and differentiating. Its integral is zero because $g$ vanishes on both plates, regardless of the forced boundary value of $\Theta$. Thus the first resonant forcing coefficient vanishes. A generic nonzero conjugate-amplitude term cannot be inferred from wave-number matching alone.

A simple eigenvalue crossing the imaginary axis signals loss of hyperbolicity and a possible local bifurcation, while the remaining eigenvalues stay away from it.

A saddle-node bifurcation occurs when a stable and an unstable equilibrium coalesce at a nonhyperbolic equilibrium and disappear as a parameter crosses a critical value.

The supercritical pitchfork normal form is $\dot{x}=\mu x-x^3$: the trivial branch is stable for $\mu <0$ and two stable nonzero branches emerge for $\mu >0$.

The normal form $\dot{x}=\mu x+x^3$ has a stable trivial equilibrium for $\mu <0$ and two unstable nonzero equilibria for $\mu <0$. They collide with the trivial branch at $\mu =0$, after which that branch is unstable.

Reflection symmetry $x\mapsto -x$ forces the reduced vector field to be odd in $x$, naturally producing paired nonzero equilibrium branches.

Adding a small symmetry-breaking term to a pitchfork bifurcation disconnects its two symmetric daughter branches and generically leaves one or more saddle-node bifurcations. This unfolding is called an imperfect pitchfork.

A bifurcation is structurally stable within a specified class of systems when every sufficiently small allowed perturbation preserves its local qualitative bifurcation diagram up to smooth changes of state and parameter coordinates. A generic saddle-node bifurcation is structurally stable in one-parameter families, whereas a pitchfork bifurcation requires a symmetry and a transcritical bifurcation requires intersecting equilibrium branches, so unrestricted perturbations destroy either one.

A bifurcation diagram plots equilibrium values against a parameter and distinguishes stable, unstable, and nonhyperbolic branch segments.

The equilibria of $\dot{x}=(a^2-x)(a-y^2)$, $\dot{y}=x-y$ lie onfor every $a$, and on $x=y=\pm \sqrt{a}$ for $a\ge 0$. They meet in a subcritical pitchfork bifurcation at $a=0$; the positive square-root branch meets the $a^2$ branch and exchanges stability in a transcritical bifurcation at $a=1$.

Forthe equilibrium branches are $x=0$, $x=\pm \sqrt{2\mu }$, and $x=\pm \sqrt{\mu -a}$ wherever the square roots are real. If $a<0$, pitchforks occur at $\mu =a$ and $\mu =0$, and the two nonzero branch pairs meet in two transcritical bifurcations at $\mu =-a$. If $a=0$, all five branches meet in one degenerate fifth-order bifurcation. If $a>0$, separate pitchforks occur at $\mu =0$ and $\mu =a$. A constant perturbation produces imperfect pitchforks and saddle nodes, so none of these unperturbed bifurcations is structurally stable without the reflection symmetry.

A Hopf bifurcation occurs when a complex-conjugate pair of eigenvalues crosses the imaginary axis and a periodic orbit is created or destroyed near the equilibrium.

The complex normal form near a simple Hopf bifurcation is $\dot{z}=(\mu +i\omega )z+\ell z|z{|}^2+\cdots$. The sign of $\mathrm{Re}\ell$ determines the critical cubic radial drift: negative gives supercritical saturation and decay at the critical parameter, while positive gives destabilizing drift.

In the radial normal forman unstable periodic orbit exists for $\mu <0$ and shrinks into the equilibrium as $\mu \uparrow 0$. This is a subcritical Hopf bifurcation.

A stable and an unstable periodic orbit coalesce into one semistable periodic orbit and disappear at a saddle-node bifurcation of periodic orbits.

Fornonzero periodic orbits haveThey are born together at $\mu =-{\lambda }^2/4$; the inner unstable orbit then vanishes in a subcritical Hopf bifurcation at $\mu =0$, while the outer orbit is stable.

In a quadrant-invariant planar system, a stable interior equilibrium branch may emerge from one boundary equilibrium and terminate at another, transferring stability at each endpoint.

Successive symmetry-breaking bifurcations can pass stability from one boundary branch to an interior branch and then to a second boundary branch.