Degeneracy Index and Poincar\'e-Hopf Theorem

TL;DR

This paper introduces a topological index for degenerate dynamical systems that extends the Poincaré-Hopf Theorem, providing insights into flow behavior near degeneracy points where the system's structure becomes singular.

Contribution

It develops a new topological index for degenerate systems, refining classical indices and capturing flow changes at singular points, extending the Poincaré-Hopf framework.

Findings

Defines a topological index for degeneracy points

Shows the index extends the Poincaré-Hopf Theorem

Provides a framework for analyzing flow changes near singularities

Abstract

A degenerate dynamical system is characterized by a state-dependent multiplier of the time derivative of the state in the time evolution equation. It can give rise to Hamiltonian systems whose symplectic structure possesses a non-constant rank throughout the phase space. Around points where the multiplier becomes singular, flow can experience abrupt and irreversible changes. We introduce a topological index for degenerate dynamical systems around these {\it degeneracy points} and show that it refines and extends the usual topological index in accordance with the Poincar\'e-Hopf Theorem.