Dynamical systems reduction through approximate lumping techniques

Abstract

Model reduction is a fundamental technique utilized across various disciplines, such as engineering, physics, and compu- tational sciences, to simplify complex mathematical models while retaining essential dynamics. This thesis introduces two novel approaches for model reduc- tion, particularly focusing on dynamical systems described by polynomial ordinary differential equations (ODEs). The pro- posed techniques aim to reduce ODE systems while providing formal error bounds for the resultant reduced models. The first approach, based on backward and forward differen- tial equivalence (BDE/FDE), partitions the set of variables in an ODE system to construct a reduced model, incorporating a tolerance parameter ε to capture perturbations in polynomial coefficients. In the second approach, we present an algorithm to transform an ODE system into a so-called differential hull. This is a construction whereby variables with structurally sim- ilar dynamics but originally different parameters may be rep- resented by the same lower and upper bounds and reduced through the backward differential equivalence. Furthermore, the thesis explores the application of these tech- niques in discovering regular equivalences on networks. An iterative scheme, called iterative ε-BDE, is introduced to com- pute regular equivalences, allowing for the analysis of roles in networks. Experimental evaluations demonstrate the effectiveness and efficiency of the proposed approaches compared to existing methods in the literature