In a zero-Hopf bifurcation, a steady state loses stability as two eigenvalues cross the imaginary axis. The unperturbed system has heteroclinic connections — orbits linking different equilibria. Under perturbation, these connections typically break: the stable and unstable manifolds separate by some distance, the “splitting.”
The splitting is exponentially small in the perturbation parameter epsilon. This makes it invisible to every order of perturbation theory — no finite Taylor expansion can detect it. Classical methods struggle because the phenomenon lives beyond all algebraic orders.
Kristiansen (arXiv:2603.12115) handles this geometrically, working entirely in the complexified phase space. The key result: the splitting magnitude is determined by the blowup time of unbounded solutions in imaginary time. Follow the heteroclinic orbit into the complex time plane. At some complex time, the solution blows up — hits a singularity. The imaginary part of that blowup time controls the exponential rate of the splitting.
The approach avoids explicit time-parameterizations entirely. Instead of tracking where orbits go as a function of time, it works with the geometry of invariant manifolds in the complexified phase space. The splitting is read off from the non-analyticity of center-like invariant manifolds near generalized saddle-nodes.
The structural lesson: a quantity that is invisible in real analysis becomes computable in complex analysis, and the computation reduces to finding a singularity. The splitting between manifolds in real space is governed by a blowup in imaginary time — the real dynamics encodes its own sensitivity in a complex-time singularity that no amount of real perturbation theory can access.