Classification of Codimension-1 Singular Bifurcations in Low-dimensional DAEs

TL;DR

This paper provides a comprehensive classification of codimension-one bifurcations in low-dimensional differential-algebraic equations, including new insights into singularity-induced and homoclinic/heteroclinic bifurcations.

Contribution

It offers the first complete list of such bifurcations for quasilinear DAEs with singularities, including normal forms and non-degeneracy conditions.

Findings

Full classification of codimension-one bifurcations in low-dimensional DAEs.

Construction of normal forms and qualitative dynamics descriptions.

Introduction of singular homoclinic and heteroclinic bifurcation analysis.

Abstract

The study of bifurcations of differential-algebraic equations (DAEs) is the topic of interest for many applied sciences, such as electrical engineering, robotics, etc. While some of them were investigated already, the full classification of such bifurcations has not been done yet. In this paper, we consider bifurcations of quasilinear DAEs with a singularity and provide a full list of all codimension-one bifurcations in lower-dimensional cases. Among others, it includes singularity-induced bifurcations (SIBs), which occur when an equilibrium branch intersects a singular manifold causing certain eigenvalues of the linearized problem to diverge to infinity. For these and other bifurcations, we construct the normal forms, establish the non-degeneracy conditions and give a qualitative description of the dynamics. Also, we study singular homoclinic and heteroclinic bifurcations, which were not considered before.