On transformation to the singularly perturbed system

Abstract

A rapid progress in hard-and software development of computational facilities as well as in numerical methods has increased the role of numerical simulations in the quantitative system analysis of many engineering problems. At the same time, the system complexity (in terms of dimensionality and non-linearity) has grown considerably increasing demand for automatic methods of analysis of qualitative system behavior. For instance, nowadays, definition of key system parameters controlling the system dynamics and finding critical regimes automatically have become crucial issue of numerical system analysis. In the present paper a transformation to the Singularly Perturbed System (SPS) as a main theoretical framework to cope with the complexity and high dimensionality will be discussed in detail. Both simple but famous and meaningful model example of Van der Pol oscillator and an example of application to numerical analysis of chemical kinetics mechanisms will be used to show the potential of the suggested framework.

Singularly perturbed profiles

“…These methods fully preserve the topological and dynamical structure of chemical reaction networks. They are based on so-called invariant manifolds of low-dimension, which define constrains (relations) between the variables of the detailed model and therefore can be efficiently used to formulate a consistent (with the overall dynamics of the detailed model) reduced model [7,8,21].…”