PSX : A Program that Explores Phase Portraits of Two-dimensional Piecewise Linear Differential Equations

Abstract

A set of all solution curves in the phase space for ordinary differential equations is called the phase portrait. Phase portraits provide global qualitative information about how dynamical systems behave under different initial conditions. In this paper, we present a program called PSX that automatically analyzes topological structure of phase portraits for two dimensional piecewise linear differential equations. PSX has several novel features that have not been achieved before : (a) PSX possesses procedures for recognizing instances of abstract concepts defined in dynamical systems theory. PSX does not only build a memory structure for instances of abstract concepts but it also directs the search process and constructively proves that what it has found is in fact an instance of the concept. (b) Though limited to two-dimensional phase spaces, PSX is applicable to complex flows in Non-Euclidean phase spaces as well as those in Euclidean phase spaces. (c) The architecture of PSX enables qualitative analysis and quantitative analysis to interact in a cooperative manner. Qualitative analysis guides the overall analysis process. When it gets stuck due to ambiguity, qualitative analysis submits a question to the quantitative analysis and the quantitative analysis provides an answer. PSX has been implemented using Common Lisp and tested against several examples.