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Special Topics in Ordinary Differential Equations: Symmetries and Reduction

The lecture will be concerned with parameter-dependent ordinary differential equations (with analytic or polynomial right-hand side), their symmetries and special types of reductions. There will be an emphasis on computational aspects and on parameters which admit symmetries, resp. reductions. For the underlying theory and for actual computations one combines algebraic and analytic results and techniques.

Block 1:
Symmetries and symmetry reductions (6 lectures; May 2 to May 11).

Basic facts and notions about analytic ordinary differential equations, including dependency theorems; solution-preserving and orbit-preserving maps, straightening theorem; tools (Lie derivative, Lie bracket, Lie series).  Infinitesimal symmetries and infinitesimal orbital symmetries of first order equations; local conditions and obstructions for symmetries; brief outlook on second order equations. Reduction of differential equations with linear (in particular toral) symmetry groups; Poincare-Dulac normal forms. (This block will be based in part on 2019 lecture notes (arXiv:1911.01053).)

Block 2:
Singular perturbations and critical parameters for chemical reaction networks (6 lectures; June 13 to June 22).

Theorems by Tikhonov and Fenichel on singular perturbations; coordinate-free characterization and reduction formulas. Application to basic biochemical reaction networks, comparison with the ``classical’’ quasi-steady state reduction in biochemistry. Identification and computation of critical parameters (``Tikhonov-Fenichel parameters’’) for singular perturbation settings; application of Gröbner bases. Some theory of chemical reaction networks (due mostly to Horn, Jackson and Feinberg); relevant notions; Deficiency Zero Theorem. Critical parameter values and reduction via structure theory of chemical reaction networks. Further examples and applications. (Notes for this block will be released in the course of the lecture.)

The written course material will be in English; the lectures will be held in English or German according to the audience’s preference.

Target audience: Students with a completed Bachelor’s degree.

Scheduled Dates


Prerequisites (referring to courses at TUM): Familiarity with the content of Analysis I-II, Lineare Algebra und Diskrete Strukturen, Gewöhnliche Differentialgleichungen is necessary. Knowledge of Algebra and Funktionentheorie is helpful.