Degeneracy Index and Poincaré-Hopf Theorem
arXiv:1907.01473
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 Poincaré-Hopf Theorem.
16 pages, 13 figures. The index definition has been polished, Lemma 3.2 and some references have been added