On partial maps derived from flows

TL;DR

This paper introduces and analyzes partial maps derived from flows, such as first-exit and first-return maps, to better understand flow dynamics and boundary behaviors, with applications to hybrid systems.

Contribution

It defines first-exit and first-return maps, generalizes related notions, and classifies boundary points, providing new tools for flow analysis and invariance results.

Findings

Boundary points classified by map behavior are topologically invariant.

Flow properties can be characterized by boundary point types.

Results apply specifically to planar flows with Jordan curve boundaries.

Abstract

The first-return map, or the Poincar\'e map, is a fundamental concept in the theory of flows. However, it can generally be defined only partially, and additional conditions are required to define it globally. Since this partiality reflects the dynamics, the flow can be described by considering the domain and behavior of such maps. In this study, we define the concept of first-exit maps and first-return maps, which are partial maps derived from flows, to enable such analysis. Moreover, we generalize some notions related to the first-return maps. It is shown that the boundary points of an open set can be classified based on the behavior of these maps, and that this classification is invariant under topological equivalence. Further, we show that some dynamical properties of a flow can be described in terms of the types of boundary points. In particular, if the flow is planar and the open set has a Jordan curve as its boundary, a more detailed analysis is possible, and we present results on the conditions which restrict possible forms of the first-exit maps . Finally, as an application of the results obtained, we consider the relationship between flows and a class of hybrid systems.