Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 65 / 2 / d

On $a+2b=1$ the coexistence fixed point hasFor $1/9<a<1$ these give a nonreal conjugate pair on the unit circle. Nearby, while the pair remains complex, its modulus is $\sqrt{D}$; increasing $a+2b$ through one changes the modulus from greater than one to less than one. This is the candidate Neimark–Sacker bifurcation, and a nonlinear coefficient is needed to decide its direction.

Here that coefficient can be calculated explicitly. Let $\rho =e^{i\theta }$, $\sigma =\sin \theta >0$, and normalize the right eigenvector and complex projection byFor deviations $(X,Y)=zq+\overline{z}\overline{q}$, the quadratic derivative isThus $f_{20}=p\mathcal{B}(q,q)/2$, $f_{11}=p\mathcal{B}(q,\overline{q})$ and $f_{02}=p\mathcal{B}(\overline{q},\overline{q})/2$ are the quadratic coefficients in $z^{\prime }$. The quadratic coordinate change hasSubstitution into the map and collection of $w^2\overline{w}$ yieldsUsing $b=(1-a)/2$ and ${\sigma }^2=(1-a)(9a-1)/(16a^2)$ simplifies the cubic radial coefficient of a quadratic planar map toConsequently, away from strong resonances of a planar map, the bifurcation is supercritical: on $a+2b>1$ coexistence is attracting, whereas on $a+2b<1$ a small attracting invariant closed curve surrounds the unstable coexistence fixed point. If $\eta =\sqrt{D}-1>0$, its leading radius is $|w{|}^2=-\eta /\ell$, and its radial multiplier is $1-2\eta +\cdots$. Motion on the curve can be quasiperiodic or frequency-locked into periodic motion; stability of the curve alone does not imply an irrational rotation.

Two interior values need qualification: $a=1/7$ is a 1:3 resonance, and $a=1/5$ is a 1:4 resonance. At the former $h_{02}$ is singular and the quadratic ${\overline{w}}^2$ term must be retained. At the latter a cubic ${\overline{w}}^3$ term remains. The resonant terms can be displayed explicitly in the same normalization. At $a=1/7$, $f_{02}=-7/6-7\sqrt{3}i/2\ne 0$. The third iterate, near the resonance and after removing nonresonant quadratics, hasHere $\delta$ is the angular detuning. Its nonzero stationary solutions have $|w|=\sqrt{{\eta }^2+{\delta }^2}/|f_{02}|$ and three phases; they constitute a saddle three-cycle. Indeed the two leading real exponents at these solutions are $3[\eta \pm \sqrt{4{\eta }^2+3{\delta }^2}]$, of opposite signs. Therefore no ordinary attracting-circle conclusion follows arbitrarily close to the exact 1:3 point.

At $a=1/5$, the cubic resonant normal form isThe extra coefficient follows from $g_{03}=f_{11}{h}_{02}+2f_{02}{\overline{h}}_{20}$ and $h=\overline{\rho }{g}_{03}$. Its radial cubic damping is uniform since $\mathrm{Re}g+|h|<0$. The slow angular equation is $\Delta \arg w=\delta +|w{|}^2\mathrm{Im}(g+h{e}^{-4i\arg w})+\cdots$, so fourfold phase locking must be considered. For example, varying $b$ at fixed $a=1/5$ gives $\delta =-\eta /2+\cdots$. The locked branches satisfy $\tan (4\arg w)=6/17$ and $|w{|}^2=-\eta /\mathrm{Re}(g+h{e}^{-4i\arg w})$. For $\eta >0$ they form one attracting and one saddle four-cycle; the two branches alternate in phase. This is a resonant alternative to quasiperiodic motion on an invariant curve. The endpoints also fail the simple-complex-pair assumptions. This distinguishes the generic nearby dynamics from unjustified claims about every point on the printed interval.