Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 78 / 2

Assume a generic saddle-focus homoclinic orbit at $\mu =0$, a smooth nondegenerate global reinjection along the selected unstable branch, and a sufficiently small neighborhood in which the linearized local flow supplies the leading passage map. Use linearizing coordinates if available; otherwise the calculation is a leading local approximation, not an exact consequence of eigenvalues alone. Choose the incoming section near $(\rho ,0,0)$ in the plane $y=0$, with coordinates $(\xi ,z)$ where $r=\rho +\xi$ and $\rho >0$. This section is transverse since $\dot{y}=\omega r\ne 0$. Choose an outgoing section $z=z_0>0$. We mark one return after a global excursion; intermediate crossings of $y=0$ during the spiral are not counted as separate global returns.

In cylindrical coordinates the local flow isFor $0<z\ll z_0$, the time to the outgoing section is $t_{\ell }={\lambda }_{+}^{-1}\log (z_0/z)$. Set $\delta =-{\lambda }_{-}/{\lambda }_{+}>0$ and $\Omega =\omega /{\lambda }_{+}$. The outgoing coordinates areThe global part of the Poincaré return map is smooth around $(X,Y)=(0,0)$ and takes the unstable-manifold point to $({\xi }^{\prime },z^{\prime })=(0,-\mu )$. Expand it aswhere the coefficients may depend smoothly on $\mu$ and the matrix is generically nonsingular. Absorbing the fixed $z_0^{-\delta }$ into them gives the leading two-dimensional return mapThe remainder refers to the smooth global-map Taylor approximation with bounded $\xi$; corrections to the local linear approximation are also neglected. The global map supplies the additive splitting parameter and the local map supplies the power and logarithmic rotation. For an iterated excursion on this same unstable branch, the next coordinate must again satisfy $z^{\prime }>0$. A nonpositive image leaves this local return domain; the formula must not be iterated through negative logarithms.

For repeated small returns, ${\xi }_{n+1}=O(z_n^{\delta })$, so $\rho +{\xi }_n$ is close to the nonzero constant $\rho$. Retain its leading value in the second component. Combining $c\cos \vartheta +d\sin \vartheta$ into a single cosine, and absorbing the section size and fixed phases, gives the Shilnikov return mapChanging the sign of the logarithmic angle changes only the phase after using cosine's evenness. A generic reinjection has $A>0$; $\Phi$ can be represented modulo $2\pi$ by a positive value. The original PDF has exponent $\delta$, which the converted TeX mistakenly replaced by $2$.

This is an asymptotic scalar approximation, not an assertion that the full map has an exact one-dimensional invariant quotient. To see why it captures the thin geometry, differentiate the two components in part (a). Their determinant is of orderwhere $C$ involves $\Omega$ and the determinant of the global linear map. For $\delta >1/2$ this tends to zero, so the return is strongly area-contracting even when its longitudinal derivative is large. Its global quadratic corrections are $O(z^{2\delta })=o(z)$ in this range. For a small fixed point, also $\xi =O(z^{\delta })$, so ignoring $\xi$ contributes that same higher order. Multipassage iterates require all passage coordinates to stay in the small domain; the leading map describes their thin folded return geometry rather than controlling arbitrary points outside it.

Let $\theta (z)=\Omega \log z+\Phi$. A fixed point of the Shilnikov return map is a periodic orbit with one marked global excursion, and satisfiesIts local passage takes logarithmically long. If the global travel time tends to $T_g$, then its physical period isEliminating $z$ and absorbing section sizes and $T_g$ into constants $C_1,C_2>0$ and a phase ${\varphi }_0$ gives the leading period–parameter relationFor the idealized leading map with a constant global travel time, the displayed two terms follow exactly; for the original flow, local and global approximation errors qualify their accuracy.

If $\delta >1$, the oscillatory term is $o(z)$ and $f_S^{\prime }(z)\to 0$. In a sufficiently small interval the map is a contraction. For $\mu <0$ there is a unique small attracting fixed point $z=-\mu +O(|\mu {|}^{\delta })$; it is an attracting periodic orbit also in the transverse direction of the full return. For $\mu \ge 0$, $f_S(z)-z=-\mu -z+O(z^{\delta })<0$ for sufficiently small positive $z$, so no periodic return orbit can remain wholly in this neighborhood. As $\mu \uparrow 0$ from below, the attracting cycle becomes the homoclinic connection andContraction also excludes higher-period cycles wholly inside that small return neighborhood. This does not exclude remote periodic orbits elsewhere in the vector field.

If $1/2<\delta <1$, the oscillatory term dominates $z$. At $\mu =0$ the fixed-point equation isThere are infinitely many positive solutions approaching zero, with phases approaching successive zeros of cosine. Their leading ratio is $z_{j+1}/z_j\to e^{-\pi /\Omega }$, so their periods satisfy $T_{j+1}-T_j\to \pi /\omega$. At these roots the sine tends to $\pm 1$, and the fixed-point multiplier has magnitude asymptotic to $A\Omega z^{\delta -1}\to \infty$. Thus these are longitudinally unstable cycles, with the thin transverse direction contracting: infinitely many saddle periodic orbits accumulate on the homoclinic connection. Expanding oscillatory lobes produce interval-covering walks and horseshoe dynamics in suitable iterates; this is the positive-saddle-value case of the Shilnikov bifurcation.

Away from the connection, the one-passage branches wiggle in the period–parameter plane. To leading order,There is no single monotone logarithmic law: the phase matters, and the exponentially smaller $-C_2{e}^{-{\lambda }_{+}T}$ term matters near cosine zeros. Folds occur when $f_S^{\prime }(z)=1$, asymptotically at $\tan \theta =\delta /\Omega$. Their parameter values alternate in sign and accumulate geometrically toward zero. On a fixed-point branch, stability is $|f_S^{\prime }(z)|<1$; small stable windows occur near stationary points of the oscillatory map. Crossing $f_S^{\prime }=-1$ gives period-doubling bifurcations, while $f_S^{\prime }=1$ gives saddle-node bifurcations of periodic orbits. Subsequent doublings and chaotic invariant sets can occur; neither all periodic orbits being stable nor a particular universal sequence for every coefficient choice is implied.

For completeness, an $m$-cycle of the return map, corresponding to a periodic orbit with $m$ global excursions, obeysand its total period isIts longitudinal multiplier is ${\prod }_j{f}_S^{\prime }(z_j)$. This is the relevant period relation for multipassage cycles; substituting their total $T$ into the one-passage cosine law would generally be incorrect.