Grammatical Description of Behaviors of Ordinary Differential Equations in Two-Dimensional Phase Space

Abstract

The major task of qualitative analysis of systems of ordinary differential equations is to recognize the global pattern of solution curves in the phase space. In this paper, I present a flow grammar, a grammatical specification of all possible patterns of solution curves one may see in the phase space. I describe flow pattern, a semi-symbolic representation of the pattern of solution curves in the phase space and show how an important class of flow patterns can be specified by the flow grammar. I then show that the flow grammar presented in this paper can generate any flow pattern resulting from any structurally stable flow on a plane. I also describe several properties of the flow grammar related to the enumeration of patterns. In particular, I estimate the upper limit of the number of application of rewriting rules needed to derive a given flow pattern. Finally, I briefly describe how the flow grammar is used in qualitative analysis to plan, monitor, and interpret the result of numerical computation.