Modelling functional dynamical systems by piecewise linear systems with delay

Abstract

Many dynamical systems in nature and technology involve delays in the interactionof variables forming the system. Furthermore, many of such systemsinvolve external inputs or perturbations which might force the system to havearbitrary initial function. The conventional way to model these systems is usingdelay differential equations (DDE). However, DDEs with arbitrary initial functionshas serious problems for finding analytical and computational solutions.This fact is a strong motivation for considering abstractions and approximationsfor dynamical systems involving delay. In this thesis, the piecewise linear systemswith delay on piecewise constant part which is a useful subclass of hybrid dynamicalsystems is studied. We introduced various representations of these systemsand studied the state transition conditions. We showed that there exists fixedpoint and periodic stable solutions. We modelled the genomic regulation of fissionyeast cell cycle. We discussed various potential uses including approximating theDDEs and finally we concluded.Keywords: piecewise linear systems, delay systems, external input, hybrid dynamicalsystems with delay, regulatory gene networks.