Analysis of intersections of trajectories of linear systems

TL;DR

This paper classifies and analyzes intersections of trajectories in linear fractional systems, providing conditions for various intersection types and generalizing separation theorems to understand system behaviors.

Contribution

It introduces a classification of trajectory intersections in linear fractional systems and generalizes the separation theorem to establish the existence of same-time intersections.

Findings

Existence of same-time intersections (EIST) proven.

Classification of systems into Type I and Type II based on EIST presence.

Necessary and sufficient conditions for nodal or cuspoidal self-intersections.

Abstract

Present article deals with trajectorial intersections in linear fractional systems ('systems'). We propose a classification of intersections of trajectories in three classes viz. trajectories intersecting at same time(EIST), trajectories intersecting at distinct times(EIDT) and self intersections of a trajectory. We prove a generalization of separation theorem for the case of linear fractional systems. This result proves existence of EIST. Based on the presence of EIST, systems are further classified in two types; Type I and Type II systems, which are analyzed further for EIDT. Besides constant solutions and limit-cycle behavior, a fractional trajectory can have nodal or cuspoidal intersections with itself. We give a necessary and sufficient condition for a trajectory to have such types of intersections.