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We start from a mechano-chemical analogy considering the time evolution of ahomogeneous chemical reaction modeled by a nonlinear dynamical system (ordinarydifferential equation, ODE) as the movement of a phase space point on thesolution manifold such as the movement of a mass point in curved spacetime.Based on our variational problem formulation \cite{Lebiedz2011} for slowinvariant manifold (SIM) computation and ideas from general relativity theorywe argue for a coordinate free analysis treatment \cite{Heiter2018} and adifferential geometry formulation in terms of geodesic flows \cite{Poppe2019}.In particular, we propose analytic continuation of the dynamical system to thecomplex time domain to reveal deeper structures and allow the application ofthe rich toolbox of Fourier and complex analysis to the SIM problem.

Analytic continuation and differential geometry views on slow manifolds and separatrices