A Diagrammatic Representation of Phase Portraits and Bifurcation Diagrams of Two-Dimensional Dynamical Systems

TL;DR

This paper introduces a diagrammatic method to systematically represent and analyze the qualitative features and bifurcations of two-dimensional dynamical systems, focusing on topological changes without quantitative details.

Contribution

It presents a novel diagrammatic approach to characterize phase portraits and bifurcations, facilitating understanding of system transitions with minimal quantitative data.

Findings

Diagrammatic representation captures all codimension 1 bifurcations.

Method allows smooth transition visualization between different phase portraits.

Guides the construction of bifurcation diagrams from partial system information.

Abstract

We treat the problem of characterizing in a systematic way the qualitative features of two-dimensional dynamical systems. To that end, we construct a representation of the topological features of phase portraits by means of diagrams that discard their quantitative information. All codimension 1 bifurcations are naturally embodied in the possible ways of transitioning smoothly between diagrams. We introduce a representation of bifurcation curves in parameter space that guides the proposition of bifurcation diagrams compatible with partial information about the system.