Degeneracy Index and Poincar\'e-Hopf Theorem

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.