# Degeneracy Index and Poincar\'e-Hopf Theorem

@article{Ruan2019DegeneracyIA, title={Degeneracy Index and Poincar\'e-Hopf Theorem}, author={Haibo Ruan and Jorge Zanelli}, journal={arXiv: Mathematical Physics}, year={2019} }

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…

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## References

SHOWING 1-10 OF 20 REFERENCES

### Degenerate dynamical systems

- Physics
- 2001

Dynamical systems, whose symplectic structure degenerates, becoming noninvertible at some points along the orbits, are analyzed. It is shown that for systems with a finite number of degrees of…

### Quantum degenerate systems

- Physics
- 2012

A degenerate dynamical system is characterized by a symplectic structure whose rank is not constant throughout phase space. Its phase space is divided into causally disconnected, nonoverlapping…

### On singular equilibria of index-1 DAEs

- Mathematics
- 2000

Some qualitative properties of singular equilibria arising in semiexplicit index-1 differential-algebraic equations are discussed in this paper. We extend a taxonomy of singularities from index-0…

### Quantum mechanics for multivalued Hamiltonians.

- MathematicsPhysical review. A, General physics
- 1987

It is shown that the path integral automatically picks up a unique combination of the branch Hamiltonians, which is a natural generalization of the Brouwer degree of the Legendre map.

### Piecewise Smooth Dynamical Systems Theory: The Case of the Missing Boundary Equilibrium Bifurcations

- Computer ScienceJ. Nonlinear Sci.
- 2016

A prototypical model is provided that can be used to generate all codimension-one boundary equilibrium collisions, and the elements of Filippov’s work that are important in achieving a full classification are summarized.

### Multiple Time Scale Dynamics

- Mathematics
- 2015

Introduction.- General Fenichel Theory.- Geometric Singular Perturbation Theory.- Normal Forms.- Direct Asymptotic Methods.- Tracking Invariant Manifolds.- The Blow-Up Method.- Singularities and…

### Dimensionally continued topological gravitation theory in Hamiltonian form

- Geology
- 1987

The most general gravitational action that yields second-order field equations in d spacetime dimensions is a sum of contributions associated with all even dimensions below d. Each contribution is…

### Classical time crystals.

- PhysicsPhysical review letters
- 2012

This work considers the possibility that classical dynamical systems display motion in their lowest-energy state, forming a time analogue of crystalline spatial order, and exhibits models of that kind, including a model with traveling density waves.

### A NORMAL FORM FOR IMPLICIT DIFFERENTIAL EQUATIONS NEAR SINGULAR POINTS

- Mathematics
- 2013

| We consider di erential equations A(x) _ x = g(x), where A is an n n-matrix of C 1 functions and g is C 1 . We investigate the above di erential equation about singular points x 0 that are standard…

### APPLIED EQUIVARIANT DEGREE, PART I: AN AXIOMATIC APPROACH TO PRIMARY DEGREE

- Mathematics
- 2006

An axiomatic approach to the primary equivariant degree
is discussed and a construction
of the primary equivariant degree via fundamental domains is presented.
For a class of equivariant maps,…