Bifurcationss of codimension one singularities of tangent vector fields on Whitney's umbrella

1988 ◽  
pp. 1-11
Author(s):  
J. Billeke ◽  
M. Wallace
Keyword(s):  
Vector Fields ◽  
Tangent Vector ◽  
Codimension One

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.

scholarly journals Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1547
Author(s):  
Stephen C. Anco ◽  
Bao Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Fields ◽  
Evolution Equations ◽  
Applied Mathematics ◽  
Tangent Vector ◽  
Solution Space ◽  
Partial Differential ◽  
Geometrical Meaning ◽  
Geometrical Formulation

A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

2014 ◽  
Vol 24 (07) ◽  
pp. 1450090 ◽  
Author(s):  
Tiago de Carvalho ◽  
Durval José Tonon
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Piecewise Smooth ◽  
Topological Equivalence ◽  
Smooth Vector ◽  
Codimension One ◽  
Piecewise Smooth Vector Fields ◽  
Piecewise Smooth Vector Field

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

scholarly journals Inverse potential problems in divergence form for measures in the plane

2021 ◽  
Author(s):  
Laurent Baratchart ◽  
Douglas Hardin ◽  
Cristobal Villalobos-Guillén
Keyword(s):  
Vector Fields ◽  
Regularization Parameter ◽  
Tangent Vector ◽  
Thin Plates ◽  
Total Variation Norm ◽  
Infinite Dimensional ◽  
Jordan Curves ◽  
Norm Minimization ◽  
Potential Problems ◽  
Noise Limit

We study inverse potential problems with source term the divergence of some unknown (R 3 -valued) measure supported in a plane; e.g., inverse magnetization problems for thin plates. We investigate methods for recovering a magnetization μ by penalizing the measure-theoretic total variation norm kμk T V , and appealing to the decomposition of divergence-free measures in the plane as superpositions of unit tangent vector fields on rectifiable Jordan curves. In particular, we prove for magnetizations supported in a plane that T V -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that T V -norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following two cases: (i) when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable; (ii) when a superset of the support is tree- like. We note that such magnetizations can be recovered via T V -regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing.

SYNCHRONY-BREAKING PERIOD-DOUBLING BIFURCATIONS FOR THREE-CELL HOMOGENEOUS COUPLED MAPS

2009 ◽  
Vol 19 (04) ◽  
pp. 1157-1167
Author(s):  
ADELA COMANICI
Keyword(s):  
Fixed Points ◽  
Network Architecture ◽  
Vector Fields ◽  
Period Doubling ◽  
Feed Forward ◽  
Codimension One ◽  
Period 2 ◽  
Coupled Networks ◽  
Coupled Maps ◽  
Period Doubling Bifurcations

Network architecture can lead to robust synchrony in coupled maps and to codimension one bifurcations from synchronous fixed-points at which the associated Jacobian is nilpotent. We discuss the codimension one synchrony-breaking period-doubling bifurcations for three-cell coupled maps. Interesting phenomena occur for all these coupled maps — a branch of period-2 points with amplitude growing as |λ|⅙ for coupled networks of feed-forward type, as well as multiple (two) branches of period-2 points with amplitude growing as |λ|½ for coupled networks of feed-forward type. We also discuss how some results related to patterns of synchrony that are valid for coupled vector fields are also valid for coupled maps.

Chaotic dynamics in ${\mathbb Z}_2$-equivariant unfoldings of codimension three singularities of vector fields in ${\mathbb R}^3$

2000 ◽  
Vol 20 (1) ◽  
pp. 85-107 ◽  
Author(s):  
FREDDY DUMORTIER ◽  
HIROSHI KOKUBU
Keyword(s):  
Vector Field ◽  
Chaotic Dynamics ◽  
Vector Fields ◽  
Equivalence Classes ◽  
Heteroclinic Cycles ◽  
Codimension One ◽  
Reasonable Possibility ◽  
Homoclinic Cycle

We study the most generic nilpotent singularity of a vector field in ${\mathbb R}^3$ which is equivariant under reflection with respect to a line, say the $z$-axis. We prove the existence of eight equivalence classes for $C^0$-equivalence, all determined by the 2-jet. We also show that in certain cases, the ${\mathbb Z}_2$-equivariant unfoldings generically contain codimension one heteroclinic cycles which are comparable to the Shil'nikov-type homoclinic cycle in non-equivariant unfoldings. The heteroclinic cycles are accompanied by infinitely many horseshoes and also have a reasonable possibility of generating suspensions of Hénon-like attractors, and even Lorenz-like attractors.

CONTROL OF CODIMENSION ONE STATIONARY BIFURCATIONS

2007 ◽  
Vol 17 (02) ◽  
pp. 575-582 ◽  
Author(s):  
FERNANDO VERDUZCO
Keyword(s):  
Control Systems ◽  
Vector Fields ◽  
Center Manifold ◽  
Control Law ◽  
Center Manifold Theory ◽  
Zero Eigenvalue ◽  
Codimension One ◽  
Pitchfork Bifurcations ◽  
Manifold Theory ◽  
Dimension One

The control of the saddle-node, transcritical and pitchfork bifurcations are analyzed in nonlinear control systems with one zero eigenvalue. It is shown that, provided some conditions on the vector fields are satisfied, it is possible to design a control law such that the bifurcation properties can be modified in some desirable way. To simplify the analysis to dimension one, the center manifold theory is used.

scholarly journals Some lie algebras of vector fields and their derivations Case of partially classical type

1981 ◽  
Vol 82 ◽  
pp. 175-207 ◽  
Author(s):  
Yukihiro Kanie
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Tangent Vector ◽  
Classical Type ◽  
Foliated Manifold

Let be a smooth foliated manifold, and the Lie algebra of all leaf-tangent vector fields on M.

ROBUST BURSTING TO THE ORIGIN: HETEROCLINIC CYCLES WITH MAXIMAL SYMMETRY EQUILIBRIA

2005 ◽  
Vol 15 (09) ◽  
pp. 2819-2832 ◽  
Author(s):  
DAVID HAWKER ◽  
PETER ASHWIN
Keyword(s):  
Limit Cycle ◽  
Vector Fields ◽  
Global Bifurcations ◽  
Heteroclinic Cycles ◽  
Maximal Symmetry ◽  
Codimension One ◽  
Local And Global Bifurcations ◽  
Resonance Bifurcation

Robust attracting heteroclinic cycles have been found in many models of dynamics with symmetries. In all previous examples, robust heteroclinic cycles appear between a number of symmetry broken equilibria. In this paper we examine the first example where there are robust attracting heteroclinic cycles that include the origin, i.e. a point with maximal symmetry. The example we study is for vector fields on ℝ3 with (ℤ2)3 symmetry. We list all possible generic (codimension one) local and global bifurcations by which this cycle can appear as an attractor; these include a resonance bifurcation from a limit cycle, direct bifurcation from a stable origin and direct bifurcation from other and more familiar robust heteroclinic cycles.

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