Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 59 / 4

For $r>0$, polar coordinates give $\dot{r}=r({\mu }_1-2x)$ and $\dot{\theta }=1$; the remaining amplitude equation is $\dot{x}={\mu }_2+x^2+r^2$. Interpret $r=0$ in the original complex coordinate, where the phase is undefined.

If ${\mu }_2<0$, put $a=\sqrt{-{\mu }_2}$. The full equilibrium points are $E_{\pm }:(z,x)=(0,\pm a)$. Their eigenvalues areThus $E_{-}$ is stable for ${\mu }_1<-2a$ and has a two-dimensional unstable manifold for ${\mu }_1>-2a$. The point $E_{+}$ has a one-dimensional unstable manifold for ${\mu }_1<2a$ and is repelling for ${\mu }_1>2a$. On ${\mu }_2=0$ they merge in a saddle-node bifurcation; the simultaneous zero and imaginary eigenvalues at $({\mu }_1,{\mu }_2)=(0,0)$ give a fold-Hopf bifurcation.

A steady positive amplitude represents a full periodic orbit, not a full equilibrium. Solving the amplitude equations givesIt exists in the wedge ${\mu }_2<-{\mu }_1^2/4$. The transverse amplitude Jacobian matrix has characteristic polynomial $s^2-{\mu }_{1}s+4r_{\ast }^2$. Its two transverse Floquet exponents have negative real part for ${\mu }_1<0$ and positive real part for ${\mu }_1>0$; the phase direction supplies the neutral Floquet multiplier one.

The wedge boundaries are Hopf bifurcations at ${\mu }_1=-2a$ from $E_{-}$ and ${\mu }_1=2a$ from $E_{+}$. The left branch is supercritical, producing a stable periodic orbit as $E_{-}$ loses stability; the right branch is subcritical in the oscillatory subspace, with an unstable periodic orbit on the side where the transverse complex pair of $E_{+}$ is stable. The real unstable direction of $E_{+}$ persists throughout. There are no full equilibrium points when ${\mu }_2>0$, and $\dot{x}>0$ then excludes a local periodic orbit. The right panel of the bifurcation diagram in part 1(b) shows the primary curves and the perturbed global curve calculated below.

Within the positive-amplitude wedge the reduced periodic orbit equilibrium has amplitude trace ${\mu }_1$ and positive determinant $4r_{\ast }^2$. At ${\mu }_1=0,{\mu }_2<0$ the transverse pair is imaginary, so the possible secondary event is a Hopf bifurcation of the amplitude equations, corresponding to an invariant torus in the full system.

It is degenerate. Set ${\mu }_2=-a^2$ and ${\mu }_1=0$. The first integral of the quadratic fold-Hopf amplitude flow isThe amplitude equations have the Hamiltonian system form $\dot{r}=F_x$, $\dot{x}=-F_r$, so $\dot{F}=0$. At $(r,x)=(a,0)$, $F$ has a strict local maximum $2a^3/3$ and negative-definite Hessian. Every nearby regular level is a closed amplitude periodic orbit, rather than one isolated orbit selected by nonlinear damping. Adding the rotating phase gives an entire family of invariant tori; the full motion is quasiperiodic or periodic according to the phase advance during one amplitude period.

For ${\mu }_1\ne 0$, the amplitude vector field has constant divergence ${\mu }_1$. The Bendixson-Dulac theorem rules out closed amplitude trajectories in the physical simply connected half-plane $r>0$. Hence there is no ordinary branch of isolated invariant tori on either side in this quadratic system. The secondary event has vanishing radial saturation coefficients; its imaginary linear pair alone does not establish a nondegenerate torus bifurcation.

With the added term, $\dot{r}=r({\mu }_1-2x+x^2)$, while $\dot{\theta }=1$ and $\dot{x}={\mu }_2+x^2+r^2$. The prescribed rescaling yieldswhere primes denote $d/d\tau$. In the limit $\epsilon =0$, the first integral is $F=-u({\lambda }_2+u^2/3+v^2)$, since $u^{\prime }=F_v$, $v^{\prime }=-F_u$.

Write ${\lambda }_2=-a^2<0$. The physical amplitude half-plane is $u\ge 0$, with $v$ of either sign; it is the radial variable $u$, not the axial variable $v$, that is nonnegative. The centre equilibrium is $(u,v)=(a,0)$, at $F=2a^3/3$. The boundary saddle equilibria are $(0,\pm a)$, both at $F=0$. For $0<F<2a^3/3$, the contours are nested closed curves around the centre. The level $F=0$ consists of the boundary $u=0$ and the interior arcOn the axis $v^{\prime }=v^2-a^2<0$, giving a heteroclinic orbit from $(0,a)$ to $(0,-a)$. On the arc $v^{\prime }=2(a^2-v^2)>0$, giving the reverse heteroclinic orbit. Together they form a heteroclinic cycle. The other parts of the zero level and negative levels give nonclosed trajectories outside this bounded region. The requested contour sketch is shown in the physical radial half-plane.

For the perturbed amplitude equations, direct differentiation of the first integral givesThe axis connection contributes zero since $u=0$. On the unperturbed interior heteroclinic orbit, $u=\sqrt{3(a^2-v^2)}$ and $v^{\prime }=2(a^2-v^2)$, so the first-order change isThe substitution $v=a\cos \varphi$ givesConsequentlyThe simple zero of this Melnikov energy-balance method calculation gives the heteroclinic splitting of a fold-Hopf amplitude cycle:This is a leading persistence curve, not an exact parameter equality for the perturbed orbit. It lies between the two primary Hopf bifurcation curves near the origin and on the negative-${\mu }_1$ side of the secondary torus curve ${\mu }_1=0$. With the added term the primary Hopf bifurcation curves themselves shift to ${\mu }_1=\pm 2\sqrt{-{\mu }_2}+{\mu }_2$.

The sign of the amplitude balance also identifies the local torus branch. Over a small closed limiting contour, $\Delta F=-\epsilon \oint u({\lambda }_1+v^2)dv$. Its enclosed-area form sets ${\lambda }_1$ equal to minus the area-average of $v^2$. This average tends to zero at the centre and to $a^2/4$ at the heteroclinic cycle. Near the centre, negative ${\lambda }_1$ attracts trajectories toward the stable amplitude equilibrium; outside a zero-balance contour, $F$ decreases and the amplitude excursion grows. Thus the secondary bifurcation creates an unstable invariant torus on the stable-periodic-orbit side. Its continuation reaches the global connection as the balance tends to ${\lambda }_1=-a^2/4$. The bifurcation diagram distinguishes this perturbed connection from the entire degenerate family present in the unperturbed system.

The normal form is invariant under $z\mapsto e^{i\phi }z$. This continuous rotational symmetry makes the axis invariant and reduces the nontrivial geometry to two amplitude variables. The axis connection from $E_{+}$ to $E_{-}$ persists without parameter tuning, while rotating the outer connection produces a family of connections from $E_{-}$ back to $E_{+}$. A two-dimensional autonomous amplitude flow cannot by itself produce the proposed saddle-focus return chaos: symmetry reduction hides the phase-dependent reinjection needed for that mechanism.

Generic higher-order terms need not retain this exact rotational symmetry. A normal form can preserve it to a finite truncation order while the full equations have small symmetry-breaking remainders. These unfold the special connection geometry into phase-dependent returns; nearby parameter curves can contain homoclinic orbits to a saddle-focus, and the Shilnikov bifurcation mechanism becomes available. Their exact placement depends on the omitted terms and is not fixed by the displayed truncation alone.

For example, at the perturbed positive-axis saddle, the real unstable eigenvalue is ${\lambda }_u=2a$ and the stable complex pair has real part ${\mu }_1-2a+a^2$. Near the calculated connection curve ${\mu }_1=-a^2/4+o(a^2)$, the contracting ratio isThis is the positive-saddle-value regime, explaining why the relevant saddle-focus return can be expanding.

In the Shilnikov return map, take a positive return coordinate $y$ and $0<\delta <1$, with nondegenerate $A\ne 0,B\ne 0$. At the connection parameter $\mu =0$, the fixed point condition isThe right side tends to zero, whereas the phase passes through infinitely many oscillations as $y\downarrow 0$. Hence there are infinitely many positive fixed points accumulating at zero, with logarithmically spaced amplitudes. For these points,whose magnitude tends to infinity along the accumulating fixed-point sequence. They represent long-period saddle periodic orbits of the full return, with period growing like ${\lambda }_u^{-1}\log (1/y)$, rather than infinitely many stable cycles.

Successive turning branches of this oscillatory return can stretch and fold across suitable intervals. The full saddle-focus Poincaré return map then admits invariant Smale horseshoes and chaotic itineraries near a generic homoclinic orbit at a Shilnikov bifurcation. Varying $\mu$ unfolds accumulating saddle-node and period-doubling events. Whether any resulting invariant set attracts trajectories depends on the other return direction and the omitted terms; the leading scalar map alone does not prove a chaotic attractor for every perturbation. Its domain also excludes negative iterates, which leave the chosen return branch. The degeneracies $A=0$ or $B=0$ would remove the oscillatory mechanism, so the conclusion uses generic, genuinely saddle-focus reinjection.