Modules of vector fields, differential forms and degenerations of differential systems

Let 𝔛r(M), r ≥ 1, denote the space of all Cr vector fields over a compact, smooth and boundaryless Riemannian manifold M of finite dimension; let [Formula: see text], 1 ≤ ℓ ≤ dim M, be the bundle of orthonormal ℓ-frames of the tangent space TM of M. For any V ∈ 𝔛r(M), Liao defined functions [Formula: see text], k = 1, …, ℓ, on [Formula: see text], which are qualitatively equivalent to the Lyapunov exponents of the differential system V. In this paper, the author shows that every [Formula: see text] depends Cr-1-continuously upon [Formula: see text] and Cr-continuously on [Formula: see text] for any given V. In addition, applying the qualitative functions, the author generalizes Liao's global linearization along a given orbit of V and considers the stochastic stability of Lyapunov spectra of linear skew-product flows based on a given ergodic system.

Differential geometry

10.1093/oso/9780198786399.003.0004 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

Differential Geometry ◽

Vector Fields ◽

Differential Forms ◽

Curvilinear Coordinates ◽

Moving Frames ◽

Tensor Fields ◽

Metric Tensor ◽

Vector Calculus ◽

Vector Version ◽

Definition Of

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

Manifolds, Vector Fields, and Differential Forms

Texts in Applied Mathematics - Introduction to Mechanics and Symmetry ◽

10.1007/978-0-387-21792-5_4 ◽

1999 ◽

pp. 121-146 ◽

Cited By ~ 2

Author(s):

Jerrold E. Marsden ◽

Tudor S. Ratiu

Keyword(s):

Vector Fields ◽

Differential Forms

Vector Fields, Differential Forms, and Derivatives

Springer Series in Synergetics - Nonlinear Waves and Solitons on Contours and Closed Surfaces ◽

10.1007/978-3-642-22895-7_4 ◽

2011 ◽

pp. 31-77

Author(s):

Andrei Ludu

Keyword(s):

Vector Fields ◽

Differential Forms

Approximation properties of harmonic vector fields and differential forms

Potential Theory - ICPT 94 ◽

10.1515/9783110818574.91 ◽

1996 ◽

pp. 91-102

Keyword(s):

Vector Fields ◽

Differential Forms ◽

Approximation Properties ◽

Harmonic Vector ◽

Harmonic Vector Fields

Limit Cycles Bifurcating from a Period Annulus in Continuous Piecewise Linear Differential Systems with Three Zones

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417500225 ◽

2017 ◽

Vol 27 (02) ◽

pp. 1750022 ◽

Cited By ~ 2

Author(s):

Maurício Firmino Silva Lima ◽

Claudio Pessoa ◽

Weber F. Pereira

Keyword(s):

Limit Cycles ◽

Vector Fields ◽

Poincaré Map ◽

Piecewise Linear ◽

Differential Systems ◽

Linear Vector ◽

Period Annulus ◽

Linear Differential Systems ◽

Linear Differential ◽

Two Parameter

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits at least two limit cycles that appear by perturbations of a period annulus. Moreover, we describe the bifurcation of the limit cycles for this class through two examples of two-parameter families of piecewise linear vector fields with three zones.

Operations on Vector Fields and Differential Forms

Universitext - Introduction to Differential Manifolds ◽

10.1007/0-387-21772-x_5 ◽

2006 ◽

pp. 105-142

Keyword(s):

Vector Fields ◽

Differential Forms

Interlude: Manifolds, Vector Fields, and Differential Forms

Texts in Applied Mathematics - Introduction to Mechanics and Symmetry ◽

10.1007/978-1-4612-2682-6_4 ◽

1994 ◽

pp. 109-129

Author(s):

Jerrold E. Marsden ◽

Tudor S. Ratiu

Keyword(s):

Vector Fields ◽

Differential Forms

LIMIT CYCLES FOR A CLASS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH THREE ZONES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412501386 ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250138 ◽

Cited By ~ 3

Author(s):

MAURÍCIO FIRMINO SILVA LIMA ◽

JAUME LLIBRE

Keyword(s):

Limit Cycle ◽

Limit Cycles ◽

Vector Fields ◽

Poincaré Map ◽

Piecewise Linear ◽

Differential Systems ◽

Linear Vector ◽

Linear Differential Systems ◽

Linear Differential ◽

Unique Limit

In this paper, we consider a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, we show that these systems admit always a unique limit cycle, which is hyperbolic.

Vector Fields and Differential Forms on the Orbit Space of a Proper Action

Axioms ◽

10.3390/axioms10020118 ◽

2021 ◽

Vol 10 (2) ◽

pp. 118

Author(s):

Larry Bates ◽

Richard Cushman ◽

Jędrzej Śniatycki

Keyword(s):

Lie Group ◽

Smooth Manifold ◽

Vector Fields ◽

Differential Forms ◽

Orbit Space ◽

Proper Action ◽

Multilinear Maps

In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space.