Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 59 / 3 / a / Solution

Take small incoming sections $y=\pm h$ and outgoing sections $x=\pm h$ after straightening the stable manifold and unstable manifold of the origin. Its eigenvalues are $1,-\delta$, so the leading local passage satisfies $x(t)=x_0{e}^t$, $y(t)=y_0{e}^{-\delta t}$. The exit time and transverse exit coordinate areThus the local passage contracts strongly as $x_0\to 0$, while its duration diverges. Smooth global reinjection from the right outgoing section to the upper incoming section has leading transverse coordinate $-\mu +K_R{y}_{\mathrm{out}}$; the corresponding left-to-lower return has $\nu +K_L{y}_{\mathrm{out}}$. Choose the parameters as these signed splitting coordinates and absorb $h^{1-\delta }$ into $A,B$.

After scaling the incoming-section labels to $y_n=\pm 1$, the leading Poincaré return map isThe second coordinate records which outgoing branch was used, not an independent continuous amplitude. A smooth planar flow preserves the orientation of a transverse-section return. For the right-to-upper pair of sections, the ordered pair consisting of the flow direction and the positive section tangent has the same orientation at both ends; the same is true for the left-to-lower pair. Hence $A,B>0$ for these sign conventions and a nondegenerate global return.

For general nonlinear $P,Q$, this is an asymptotic return model, with smooth parameter corrections and higher-order local-passage corrections. The hypotheses do not imply an exact power-law map in the original coordinates. Interpreting the printed parameters themselves as linear splitting coordinates also requires a nondegenerate two-parameter unfolding: the two splitting derivatives must be nonzero. For example, composing a generic right-loop splitting with $\mu \mapsto {\mu }^2$ preserves the stated loop at $\mu =0$ but does not give the printed signed $-\mu$ offset. The subsequent region diagram uses the standard generic splitting-coordinate interpretation. Nor is $x=0$ a normal return point: it lies on the stable manifold, whose trajectory takes infinite time to reach the saddle equilibrium. This explains how the map describes nearby recurrent trajectories without extending a finite-time return across the singular separatrix.