Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 60 / 2

Represent the leading real pattern by $A(T)e^{i{k}_{c}x}+\overline{A}(T)e^{-i{k}_{c}x}$. Before forcing, continuous translation symmetry acts as $A\mapsto e^{i\theta }A$, and spatial reflection acts as $A\mapsto \overline{A}$. Equivariance excludes quadratic terms and permits a linear term and $|A{|}^{2}A$ at cubic order. Symmetry under reflection makes their coefficients real. A supercritical bifurcation has a saturating cubic coefficient; rescaling amplitude and time makes it $-1$. The imperfection has a Fourier component at the same wavenumber, so its resonant projection contributes an additive real coefficient. On the spatially uniform wave envelope subspace the amplitude equation is therefore $A_T=\mu A-|A{|}^{2}A+\alpha$. Allowing modulations would additionally produce the spatial terms of the real Ginzburg–Landau equation.

For a forcing pattern moving in the positive $x$ direction, its laboratory-frame coefficient is $\alpha e^{-i\Omega T}$, where $\Omega$ is proportional to $k_{c}c$ after converting to slow time. The comoving amplitude is $B=e^{i\Omega T}A$, since the fast carrier becomes $e^{i{k}_c(x-ct)}$. Differentiation givesA constant phase choice makes $\alpha >0$, and complex conjugation reverses the sign of $\Omega$, so take $\Lambda =\Omega >0$. Put $B={\alpha }^{1/3}a$ and $\tau ={\alpha }^{2/3}T$. The rescaled linear coefficients are $\mu /{\alpha }^{2/3}$ and $\Lambda /{\alpha }^{2/3}$, and the forcing becomes one. Below the rescaled variables are renamed $A,T,\mu ,\Lambda$, giving the directly forced cubic amplitude equation.

Write $s=R^2$. A steady complex amplitude obeys $(\mu -s+i\Lambda )A=-1$, soThus every positive intensity root gives exactly one steady phase, and $s=0$ is impossible. The cubic intensity polynomial is $F(s,\mu )=s^3-2\mu s^2+({\mu }^2+{\Lambda }^2)s-1$. Its positive steady branch can be parameterized completely asThe two formulae join smoothly as a single curve at the maximum radius $R={\Lambda }^{-1}$; this joining point is not a saddle-node bifurcation. A turning point with respect to $\mu$ satisfies $F_s=0$. Subtracting the equilibrium and derivative conditions givesThese points lie on ${\mu }_{+}$. Since $f^{\prime }(s)=(1-s^3)/s^5$, its maximum is $f(1)=3/4$. Therefore there is exactly one distinct steady state for every $\mu$ when $\Lambda \ge \sqrt{3}/2$. One can also see uniqueness directly: ${\mu }_{-}$ is increasing and ${\mu }_{+}$ is decreasing when no turning points exist, and their respective ranges meet only at their common endpoint. At the equality there is a cusp bifurcation with a triple intensity root at $s=1,\mu =3/2$.

When $0<\Lambda <\sqrt{3}/2$, the two turning intensities satisfy $s_{-}<1<s_{+}$. Define ${\mu }_L=s_{-}+1/(2s_{-}^2)$ and ${\mu }_U=s_{+}+1/(2s_{+}^2)$; the curve has a local minimum at the former and a local maximum at the latter, with ${\mu }_L<{\mu }_U$. There are three distinct steady intensities for ${\mu }_L<\mu <{\mu }_U$, two distinct ones at an endpoint, and one otherwise. The bifurcation diagrams below show the folded small-$\Lambda$ curves and the unfurled large-$\Lambda$ curve, together with the linear stability calculated next. Both ends of the full branch have $R\to 0$ as $|\mu |\to \infty$, while its middle reaches the finite maximum ${\Lambda }^{-1}$.

Let the equilibrium be $A_{\ast }=R{e}^{i\theta }$ and write $\delta A=e^{i\theta }(u+iv)$. The linearization of the cubic term gives $\delta (|A{|}^{2}A)=2s\delta A+A_{\ast }^2\delta \overline{A}$. Therefore the real stability matrix isIts eigenvalues obeyThe square root may be imaginary; stability uses real parts. The determinant $D$ is exactly $F_s$, so a zero eigenvalue occurs only at the turning points derived above. Away from the cusp those are ordinary saddle-node bifurcations; the parameterization joining point has $D>0$.

At a candidate Hopf bifurcation, the trace vanishes: ${\mu }_H=2s_H$. Substitution into the steady cubic gives $s_H(s_H^2+{\Lambda }^2)=1$, which has one positive root. The pair is genuinely imaginary only ifThus $s_H<2^{-1/3}$ and, equivalently,This direct calculation proves the threshold without needing the additional monotonicity assumption. The equality gives a double-zero eigenvalue, not an ordinary Hopf bifurcation. Along a fixed-$\Lambda$ branch, implicit differentiation gives $ds/d\mu =-2s(\mu -s)/D$; at this crossing $d\mathrm{Re}\sigma /d\mu =1+4s_H^2/D_H>0$. A nonzero nonlinear Hopf coefficient then gives the usual Hopf bifurcation.

The fold and Hopf thresholds of a directly forced cubic amplitude divide the diagrams into three regimes. At noncritical points the two eigenvalues, counted with multiplicity by the signs of their real parts, are classified as follows:

At a fold one eigenvalue is zero; at a Hopf point the pair has zero real part. These nonhyperbolic points are marked separately in the sketches.