Quasi periodic flow near a codimension one singularity of a divergence free vector field in dimension three

1981 ◽  
pp. 75-89 ◽  
Author(s):  
Henk Broer
Keyword(s):  
Vector Field ◽  
Periodic Flow ◽  
Codimension One ◽  
Divergence Free Vector ◽  
Divergence Free

Analyticity and metric transitivity on the torus

1998 ◽  
Vol 18 (3) ◽  
pp. 717-723
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Differential Equations ◽  
Entire Plane ◽  
Transitive Flow ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Open Question ◽  
Set Of Points

Suppose we are given an analytic divergence free vector field $(X,Y)$ on the standard torus. We can find constants $a$ and $b$ and a function $F(x,y)$ of period one in both $x$ and $y$ such that $(X,Y)=(a-F_y,b+F_x)$. For a given $F$, let $P$ be the map sending $(x,y)$ into $(F_y(x,y),-F_x(x,y))$. Let $A$ be the image of the torus under this map and let $B$ be the image under this map of the set of points $(x,y)$ at which $F_{xx}F_{yy}-(F_{xy})^2$ vanishes. For any point $(a,b)$ in the complement of the interior of $A$, the flow on the torus arising from the differential equations $dx/dt=a-F_y(x,y)$, $dy/dt=b+F_x(x,y)$ is metrically transitive if and only if $a/b$ is irrational. For any point in $A$ but not in $B$ the flow is not metrically transitive. Moreover, if $a/b$ is irrational but the flow on the torus is not metrically transitive and we use our differential equations to define a flow in the entire plane (rather than on the torus), this flow has a nonstationary periodic orbit. It is an open question whether a point $(a,b)$ in the interior of $A$ can give rise to a metrically transitive flow.

scholarly journals Divergence-free vector-field and reduction

1998 ◽  
Vol 244 (5) ◽  
pp. 377-382 ◽  
Author(s):  
Debin Huang ◽  
Xiaohua Zhao ◽  
Zengrong Liu
Keyword(s):  
Vector Field ◽  
Divergence Free Vector ◽  
Divergence Free

A divergence-free vector field model for imaging applications

2009 ◽  
Author(s):  
O. Skrinjar ◽  
A. Bistoquet ◽  
J. Oshinski ◽  
K. Sundareswaran ◽  
D. Frakes ◽  
...  
Keyword(s):  
Vector Field ◽  
Field Model ◽  
Divergence Free Vector ◽  
Divergence Free

scholarly journals Vector fields satisfying the barycenter property

2018 ◽  
Vol 16 (1) ◽  
pp. 429-436 ◽  
Author(s):  
Manseob Lee
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Axiom A ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe show that if a vector fieldXhas theC1robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a genericC1-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].

Basic Technical Outline

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Vector Field ◽  
Bounded Number ◽  
Divergence Free Vector ◽  
Individual Wave ◽  
Error Terms ◽  
Interaction Term ◽  
Divergence Free ◽  
Complex Valued ◽  
Stress Term ◽  
Transport Term

This chapter provides a more technical outline of the construction, starting with a solution to the Euler-Reynolds system and a correction v₁ = v + V, p₁ = p + P. The correction V is a divergence free vector field which oscillates rapidly compared to v. In the construction, there will always be a bounded number of waves Vsubscript I (at most 192) which are nonzero at any given time t. Each individual wave Vsubscript I composing V is a complex-valued, divergence free vector field that oscillates rapidly in only one direction. The chapter introduces several ways in which to represent each Vsubscript I. Finally, it presents five main error terms: the Transport term, the High–Low Interaction term, the High–High Interference terms, the Stress term, and the Mollification terms.

A pasting lemma and some applications for conservative systems

2007 ◽  
Vol 27 (5) ◽  
pp. 1399-1417 ◽  
Author(s):  
ALEXANDER ARBIETO ◽  
CARLOS MATHEUS
Keyword(s):  
Vector Field ◽  
Compact Manifold ◽  
Vector Fields ◽  
Three Dimensional ◽  
Invariant Set ◽  
Dominated Splitting ◽  
Divergence Free Vector ◽  
Conservative Systems ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

scholarly journals Representation of vector fields

2015 ◽  
Vol 3 (2) ◽  
pp. 73
Author(s):  
Alexander G. Ramm
Keyword(s):  
Vector Field ◽  
Explicit Formula ◽  
Simple Proof ◽  
Vector Fields ◽  
Smooth Domain ◽  
Gradient Field ◽  
Smooth Vector ◽  
Smooth Vector Field ◽  
Divergence Free Vector ◽  
Divergence Free

<p>A simple proof is given for the explicit formula which allows one to recover a \(C^2\) – smooth vector field \(A=A(x)\) in \(\mathbb{R}^3\), decaying at infinity, from the knowledge of its \(\nabla \times A\) and \(\nabla \cdot A\). The representation of \(A\) as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded \(C^2\) - smooth domain.</p>

scholarly journals On the structure of the family of Cherry fields on the torus

1985 ◽  
Vol 5 (1) ◽  
pp. 27-46 ◽  
Author(s):  
Colin Boyd
Keyword(s):  
Vector Field ◽  
Hausdorff Dimension ◽  
Rotation Number ◽  
Vector Fields ◽  
Cantor Set ◽  
Irrational Rotation ◽  
Codimension One ◽  
The Family ◽  
Irrational Rotation Number

AbstractA class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure ofhas irrational rotation number}, where ƒ is a certain map of the circle andRtis rotation throught. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.

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