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.

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.

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.

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.

scholarly journals Perturbation of the hope flow and transverse foliations

1991 ◽  
Vol 122 ◽  
pp. 75-82 ◽  
Author(s):  
Shigenori Matsumoto ◽  
Atsushi Sato
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Closed Manifold ◽  
Codimension One ◽  
Plane Field

Consider a nonsingular vector field X on a closed manifold Mn. As a matter of fact, X always admits a transverse codimension one plane field, which however may fail to be integrable. In fact it is well known that there are many examples of vector fields which do not admit transverse foliations.

scholarly journals Pairings between bounded divergence-measure vector fields and BV functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Graziano Crasta ◽  
Virginia De Cicco ◽  
Annalisa Malusa
Keyword(s):  
Vector Field ◽  
Conservation Laws ◽  
Bounded Variation ◽  
Vector Fields ◽  
Hyperbolic Conservation Laws ◽  
Divergence Measure ◽  
Compensated Compactness ◽  
Hyperbolic Conservation ◽  
Bv Functions ◽  
Bounded Functions

AbstractWe introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in \mathrm{BV}. We remark that the standard pairing in general does not share this property.

scholarly journals The geometry of null-like disformal transformations

2019 ◽  
Vol 16 (11) ◽  
pp. 1950180 ◽  
Author(s):  
I. P. Lobo ◽  
G. G. Carvalho
Keyword(s):  
Vector Field ◽  
Conformal Transformation ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Spinor Structure ◽  
Schwarzschild Metric ◽  
Killing Vectors ◽  
Symmetry Properties ◽  
Geometric Objects

Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.

Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems

1991 ◽  
Vol 11 (3) ◽  
pp. 443-454 ◽  
Author(s):  
Morris W. Hirsch
Keyword(s):  
Vector Field ◽  
Positive Integer ◽  
Differential Equations ◽  
Vector Fields ◽  
3 Dimensional ◽  
Planar Surfaces ◽  
Systems Of Differential Equations ◽  
Nonwandering Point

AbstractFor certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.

SINGULAR CYCLES AND Ck-ROBUST TRANSITIVE SET ON MANIFOLD WITH BOUNDARY

2011 ◽  
Vol 13 (02) ◽  
pp. 191-211 ◽  
Author(s):  
D. CARRASCO-OLIVERA ◽  
C. A. MORALES ◽  
B. SAN MARTÍN
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Manifold With Boundary ◽  
Transitive Set ◽  
Singular Cycle ◽  
Hyperbolic Equilibrium

Let M be a 3-manifold with boundary ∂M. Let X be a C∞, vector field on M, tangent to ∂M, exhibiting a singular cycle associated to a hyperbolic equilibrium σ∈∂M with real eigenvalues λss < λs < 0 < λu satisfying λs - λss - 2λu > 0. We prove under generic conditions and k large enough the existence of a Ck robust transitive set of X, that is, any Ck vector field Ck close to X exhibits a transitive set containing the cycle. In particular, C∞ vector fields exhibiting Ck robust transitive sets, for k large enough, which are not singular-hyperbolic do exist on any compact 3-manifold with boundary.

FINDING PARAMETERS FOR CHUA’S OSCILLATOR WHICH IS TOPOLOGICALLY CONJUGATE TO SYSTEMS WITH A SCALAR NONLINEARITY

1995 ◽  
Vol 05 (03) ◽  
pp. 895-899 ◽  
Author(s):  
CHAI WAH WU ◽  
LEON O. CHUA
Keyword(s):  
Vector Field ◽  
Large Class ◽  
Vector Fields ◽  
Colpitts Oscillator ◽  
Chua’S Oscillator ◽  
Scalar Nonlinearity

Chua’s oscillator is topologically conjugate to a large class of vector fields with a scalar non-linearity. In this letter, we give an algorithm which, given a vector field in this class, finds the parameters for Chua’s oscillator for which Chua’s oscillator is topologically conjugate to it. We illustrate this by transforming Sparrow’s system and the chaotic Colpitts oscillator into equivalent Chua’s oscillators.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

2021 ◽  
Vol 62 ◽  
pp. 53-66
Author(s):  
Fethi Latti ◽  
◽  
Hichem Elhendi ◽  
Lakehal Belarbi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Harmonic Vector ◽  
New Class ◽  
Necessary And Sufficient ◽  
Sasakian Metric ◽  
Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

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