Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 56 / 3

A global bifurcation joins two mechanisms: a long passage near a hyperbolic equilibrium point and a finite excursion returning the unstable manifold to its neighborhood. Eigenvalue ratios control the local passage, while the geometry of the global return decides reinjection, folding and trapping. In the real symmetric case there are two returning branches and a power-law cusp. In the saddle-focus case rotation adds infinitely many oscillations to that cusp. The following maps display both mechanisms explicitly.

For the Lorenz configuration, enter on $z=h$ at $(x,y,h)$ and leave on $x=sh$, where $s=\mathrm{sgn}(x)$. Solving the linear flow gives a flight timePut $\gamma =-\hat{\lambda }/{\lambda }_{+}>\delta =-{\lambda }_{-}/{\lambda }_{+}$. The outgoing coordinates areThe smooth global map from each outgoing section to the incoming section has, to leading order,up to the choice of which unstable branch carries the sign of $\nu$. Reflection symmetry makes the offsets opposite and the leading coefficients equal. Since the stable transverse coordinate is multiplied by the higher power $|x{|}^{\gamma }$, it contributes only subleading terms to the leading scalar map. Absorbing powers of $h$ into a positive constant gives the Lorenz power return mapThe line $x=0$ lies on the stable manifold and has no finite return; the discontinuity is therefore a genuine missing return, not an arbitrary value to assign at zero.

If $\delta >1$, the derivative $A\delta |x{|}^{\delta -1}$ tends to zero. For $\mu <0$, there are two attracting fixed points, $x=\pm s$ with $s=-\mu +A{s}^{\delta }\simeq -\mu$. They describe two symmetry-related one-lobe attracting flow periodic orbits. For $\mu >0$, the orbit alternates between $s$ and $-s$, where $s=\mu -A{s}^{\delta }\simeq \mu$; its multiplier is $(A\delta s^{\delta -1}{)}^2<1$. This represents one attracting flow periodic orbit traversing both lobes. At $\mu =0$ the two homoclinic loops meet: this is the contracting gluing scenario, with diverging flow period at the connection.

If $\delta <1$, the local derivative is unbounded, and the small periodic orbits are repelling in the expanding return direction. For $\mu >0$, the small positive fixed point has $s\simeq (\mu /A{)}^{1/\delta }$, with a multiplier tending to infinity; there is a symmetry-related negative fixed point as well. More complex motion follows from the two branches, not merely from a large derivative at one fixed point. Choose $I=[-c\mu ,c\mu ]$ with $0<c<1$. The inverse branches areFor sufficiently small positive $\mu$, both lie inside $I$, their images are disjoint, and their derivatives are $O({\mu }^{1/\delta -1})\to 0$. Nested inverse images for every infinite sign sequence therefore define a Cantor invariant set. Iterating the forward map shifts the sequence. This constructs periodic itineraries of every symbolic period and nonperiodic itineraries; transverse contraction in the full return turns the expanding two-strip mechanism into a horseshoe.

This is the local chaotic Lorenz mechanism. A robust attracting Lorenz set additionally needs a global trapping region and the appropriate return properties. The scalar asymptote and $\delta <1$ alone do not prove that every nearby orbit is attracted to one chaotic set; periodic windows, escaping orbits and other global returns must be distinguished. Symmetry is crucial to obtaining the two homoclinic branches together in a single parameter variation.

In cylindrical coordinates the saddle-focus linearization isUse a small incoming neighborhood of the stated section point $(\rho ,0,-\mu )$. A nearby incoming cylinder $r=\rho$ is smoothly equivalent to that section for this calculation. Denote the positive unstable passage coordinate by $x=z_{\mathrm{in}}>0$ and leave at $z=h$. ThenThe global map is smooth in the outgoing Cartesian coordinates $r_{\mathrm{out}}\cos {\theta }_{\mathrm{out}}$ and $r_{\mathrm{out}}\sin {\theta }_{\mathrm{out}}$. Its returned unstable coordinate therefore has the leading formCombining the two trigonometric terms, absorbing the fixed scales into the amplitude and phase, and using the evenness of cosine givesThe returned radial section coordinate is $\rho +O(x^{\delta })$, supplying the thin transverse direction of the full two-dimensional return. In particular its determinant has order $x^{2\delta -1}$; the exponent is also obtained by integrating the flow divergence ${\lambda }_{+}+2{\lambda }_{-}$ over the local flight. Thus $\delta >1/2$ is the volume-contracting range. Corrections from the returned radial deviation enter at order $x^{2\delta }$, smaller than the fixed-point term $x$ in this range. Only positive returned $x$ on the chosen branch belong to the scalar return domain.

For $\delta >1$,For small $\mu <0$, there is one small attracting fixed point $x=-\mu +O(|\mu {|}^{\delta })$, representing a stable long-period periodic orbit. It ends in the homoclinic connection at $\mu =0$; for $\mu >0$ there is no sufficiently small positive fixed point. The oscillations alone do not create expanding chaos when their derivative tends to zero.

For $1/2<\delta <1$, there is both net volume contraction and a positive saddle equilibrium value ${\lambda }_{+}+{\lambda }_{-}>0$. At the connection, fixed points satisfyThe right side tends to zero, while the phase runs through infinitely many rotations. There are therefore infinitely many positive fixed points accumulating at zero near successive cosine zeros. Their successive size ratios tend to $e^{-\pi /q}$ and their derivative magnitudes diverge. In the full return they are saddle equilibrium periodic orbits: one multiplier expands and the thin transverse direction contracts. Their flow periods grow as ${\lambda }_{+}^{-1}\log (h/x)$.

The oscillatory graph also supplies the complicated itineraries. Take a sufficiently small positive return interval. Near successive zero crossings, narrow monotone subintervals map across that interval with arbitrarily large slope. Two or more inverse branches contract into it, so repeated inverse choices give symbolic invariant sets just as in part (a); lifting the strips to the thin two-dimensional return gives horseshoes and infinitely many periodic orbits. This explains the Shilnikov chaotic mechanism rather than only naming it.

For nonzero splitting, the graph shifts vertically. Fixed-point tangencies with slope $+1$ create saddle-node bifurcations of periodic orbits; crossings of slope $-1$ give period doubling. Near an oscillatory extremum the derivative can be small, allowing stable periodic windows. Such thresholds accumulate geometrically toward the homoclinic parameter, interspersed with expanding returns. Hence the unfolding can contain complicated switching, arbitrarily long periods and chaos, rather than a single attracting periodic branch. This is log-periodic accumulation of Shilnikov cycles. The conclusion concerns invariant sets and possible attractors allowed by the return geometry; it does not assert that a homoclinic connection alone makes all nearby trajectories chaotic.