Qualitative analysis of behavior of systems of piecewise linear differential equations with two state variables

Abstract

The set of all solution curves in the phase space for a system of ordinary differential equations (ODEs) is called the phase portrait. A phase portrait provides global qualitative information about how a given system of ODEs behaves under different initial conditions. We are developing a method called Topological Flow Analysis (TFA) for automating analysis of the topological structure of the phase portrait of systems of ODEs. In this paper, we describe the first version of TFA for systems of piecewise linear ODEs with two state variables. TFA has several novel features that have not been achieved before. Firstly, TFA enables to grasp characteristics of all behaviors of a given system of ODEs. Secondly, TFA allows to symbolically represent behaviors in terms of critical geometric features in the phase space. Finally, TFA integrates qualitative and quantitative analysis. The current version of TFA has been implemented as a program called PSX2PWL using Common Lisp.