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Introduction to the Geometric Theory of ODEs with Applications to Chemical Processes

  • Dietrich FlockerziEmail author
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

We give an introduction to the geometric theory of ordinary differential equations (ODEs) tailored to applications to biochemical reaction networks and chemical separation processes. Quite often, the ordinary differential equations under investigation are “reduced” partial differential equations (PDEs) as in the search of traveling wave solutions. So, we also address ODE topics that have their origin in the PDE context.

We present the mathematical theory of invariant and integral manifolds, in particular, of center and slow manifolds, which reflect the splitting of variables and/or processes into slow and fast ones. The invariance of a smooth manifold is characterized by a quasilinear partial differential equation, and the widely used approximations of invariant manifolds are derived from such PDEs. So we also offer, to some extent, an introduction to quasilinear PDEs. The basic ideas and crucial tools are illustrated with numerous examples and exercises. Concerning the proofs, we confine ourselves to outline the crucial steps and refer, especially in the first three sections, to the literature.

The final Sects. 1.4 and 1.5 on reaction–separation processes and on chromatographic separation present new results, including their proofs. They are the outcome of many fruitful discussions with my colleagues Malte Kaspereit and Achim Kienle.

Keywords

Stability Integral manifolds and method of characteristics Center manifolds and asymptotic phases Reduction methods and bifurcations Quasi-stationary approximations and singular perturbations Slow invariant manifolds Reactive and chromatographic separation networks 

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

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