Singularly Perturbed Dynamical Systems

Final Report Summary - SINGPERTDYNSYS (Singularly Perturbed Dynamical Systems)

The central object of study in the project are so-called singularly-perturbed or fast-slow dynamical systems. Typical examples arise in chemical as well as many biological systems. For example, the reaction rates in a multi-molecule chemical reaction often differ by several orders of magnitude and this tends to lead to differential equations with small parameters. Frequently, the small parameters imply a time-scale separation between different dynamical variables and therefore one has to deal with so-called fast-slow dynamical systems. A basic goal of the project was to advance the mathematical theory and numerical treatment of such fast-slow systems in several respects. The first goal of the project was to understand oscillatory patterns. In this regard, several frontiers were tackled and progress has been made to understand mechanisms for autocatalytic reaction models better, to solve a long-standing open problem in the context of fast-slow models for the peroxidase-oxidase reaction, and to investigate the interplay between stochastic perturbations and oscillation patterns. The significant advances are that we now understand the role of certain small parameters, and the interplay between several small parameters a lot better. For the stochastic case, it should be highlighted that the new results also led to many unexpected applications related to a newly emerging theory of early-warning signs. A related second objective of the project was to understand multi-parameter problems better, particularly using a mathematical geometric desingularization technique (also called blow-up method). Roughly speaking, this method allows the use of linear theory where one usually would expect to have to use nonlinear methods. In this context, several advances in the project were made, particularly in higher-dimensional problems as well as in reaching a better understanding of the interplay between small noise and small time-scale separation. The last key goal of the project was to extend the ideas from ordinary differential equations (i.e. purely temporal problems) to partial differential equations (i.e. usually space-time problems) in certain special cases. This has been achieved in several regards developing efficient numerical tools for singularly-perturbed reaction-diffusion systems as well as new algorithms for problems with small noise. Furthermore, the project has even succeeded to apply some ideas to stochastic partial differential equations exhibiting wave propagation phenomena as well as to certain classes of nonlocal equations with small parameters. Overall, the project has led to new tools and mathematical methods that are widely applicable in the natural sciences and engineering and several direct applications have also been carried out in the project in transdisciplinary collaborations.

Regarding the career perspectives and re-integration of the fellow (Christian Kuehn), he has now submitted his habilitation. Furthermore, he has obtained a highly competitive Lichtenberg-Professorship from the Volkswagen Foundation (only three awards made across all sciences in Germany in 2015) which is due to start in 2016. It is expected to that the Lichtenberg-Professorship also leads to a tenure-track position starting in 2016.

Final Report Summary - SINGPERTDYNSYS (Singularly Perturbed Dynamical Systems)

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