The mathematics of processes with fast and slow stages
The SINGPERTDYNSYS (Singularly perturbed dynamical systems) project focused on so-called singularly-perturbed or fast-slow dynamical systems such as the reaction rates in a multi-molecule chemical reaction. Such systems often have dynamical variables that evolve on different timescales. The first goal of the project was to understand oscillatory patterns. Progress was made with regard to understanding mechanisms for autocatalytic reaction models. This should help researchers solve a long-standing open problem in the context of fast-slow models for the peroxidase-oxidase reaction. Furthermore, this is also applicable for the interplay between stochastic perturbations and oscillation patterns. The role of small parameters, and the interplay between parameters was investigated. For the stochastic case, the new results have led to many unexpected applications in the newly emerging theory of early-warning signs. A related second objective of the project was to understand multi-parameter problems better, using a mathematical geometric desingularisation technique (also called the blow-up method). This method allows the use of linear theory instead of nonlinear methods. Particular advances were made in higher-dimensional problems and 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 special cases. This was achieved by developing efficient numerical tools for singularly perturbed reaction-diffusion systems, and new algorithms for problems with small noise. The project managed 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. New tools and mathematical methods have been discovered that are widely applicable in the natural sciences and engineering. Several direct applications have already been provided in transdisciplinary collaborations.
Keywords
Fast-slow system, singularly perturbed system, oscillatory pattern, desingularisation technique, blow-up method, differential equation
