# Analytic continuation and differential geometry views on slow manifolds and separatrices

## TL;DR

This paper introduces a differential geometry approach to analyzing slow manifolds and separatrices in dynamical systems, utilizing analytic continuation to complex time for deeper structural insights.

## Contribution

It proposes a coordinate-free, differential geometry framework for slow manifold computation, incorporating analytic continuation to complex time to enhance analysis capabilities.

## Key findings

- Analytic continuation reveals deeper structures in dynamical systems.
- Differential geometry provides a coordinate-free analysis of slow manifolds.
- The approach connects variational principles with complex analysis tools.

## Abstract

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