Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields

2006 ◽  
pp. 636-686
Keyword(s):  
Fixed Points ◽  
Vector Fields ◽  
Three Dimensional ◽  
Hyperbolic Fixed Points

scholarly journals Emergent chiral symmetry in a three-dimensional interacting Dirac liquid

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
András L. Szabó ◽  
Bitan Roy
Keyword(s):  
Fixed Points ◽  
Chiral Symmetry ◽  
Rotational Symmetry ◽  
Three Dimensional ◽  
Spatial Dimension ◽  
Dirac Fermions ◽  
Electronic Interactions ◽  
Emergent Phenomena ◽  
Ordered Phases ◽  
The Stability

Abstract We compute the effects of strong Hubbardlike local electronic interactions on three-dimensional four-component massless Dirac fermions, which in a noninteracting system possess a microscopic global U(1) ⊗ SU(2) chiral symmetry. A concrete lattice realization of such chiral Dirac excitations is presented, and the role of electron-electron interactions is studied by performing a field theoretic renormalization group (RG) analysis, controlled by a small parameter ϵ with ϵ = d−1, about the lower-critical one spatial dimension. Besides the noninteracting Gaussian fixed point, the system supports four quantum critical and four bicritical points at nonvanishing interaction couplings ∼ ϵ. Even though the chiral symmetry is absent in the interacting model, it gets restored (either partially or fully) at various RG fixed points as emergent phenomena. A representative cut of the global phase diagram displays a confluence of scalar and pseudoscalar excitonic and superconducting (such as the s-wave and p-wave) mass ordered phases, manifesting restoration of (a) chiral U(1) symmetry between two excitonic masses for repulsive interactions and (b) pseudospin SU(2) symmetry between scalar or pseudoscalar excitonic and superconducting masses for attractive interactions. Finally, we perturbatively study the effects of weak rotational symmetry breaking on the stability of various RG fixed points.

scholarly journals $$ \mathcal{N} $$ = 1 QED in 2 + 1 dimensions: dualities and enhanced symmetries

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Francesco Benini ◽  
Sergio Benvenuti
Keyword(s):  
Fixed Points ◽  
Three Dimensional ◽  
Global Symmetry ◽  
Mass Terms

Abstract We consider three-dimensional sQED with 2 flavors and minimal supersymmetry. We discuss various models which are dual to Gross-Neveu-Yukawa theories. The U(2) ultraviolet global symmetry is often enhanced in the infrared, for instance to O(4) or SU(3). This is analogous to the conjectured behaviour of non-supersymmetric QED with 2 flavors. A perturbative analysis of the Gross-Neveu-Yukawa models in the D = 4 − ε expansion shows that the U(2) preserving superpotential deformations of the sQED (mod- ulo tuning mass terms to zero) are irrelevant, therefore the fixed points with enhanced symmetry are stable. We also construct an example of $$ \mathcal{N} $$ N = 2 sQED with 4 flavors that exhibits enhanced SO(6) symmetry.

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

Reconstructing Three-Dimensional Fluid Velocity Vector Fields from Acoustic Transmission Measurements

1977 ◽  
pp. 307-326 ◽  
Author(s):  
S. A. Johnson ◽  
J. F. Greenleaf ◽  
C. R. Hansen ◽  
W. F. Samayoa ◽  
M. Tanaka ◽  
...  
Keyword(s):  
Velocity Vector ◽  
Vector Fields ◽  
Fluid Velocity ◽  
Three Dimensional ◽  
Acoustic Transmission ◽  
Transmission Measurements

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Mircea Crasmareanu ◽  
Camelia Frigioiu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Conformal Killing ◽  
Space Forms ◽  
Potential Vector ◽  
Legendre Curves ◽  
Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

2018 ◽  
Vol 28 (11) ◽  
pp. 1850139 ◽  
Author(s):  
Laigang Guo ◽  
Pei Yu ◽  
Yufu Chen
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Center Manifold ◽  
Three Dimensional ◽  
Center Manifold Theory ◽  
Normal Form Theory ◽  
Quadratic Vector Fields ◽  
Fine Focus ◽  
Form Theory ◽  
Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

scholarly journals On the symmetries of three-dimensional generalized symmetric spaces

2021 ◽  
Vol 13(62) (2) ◽  
pp. 451-462
Author(s):  
Lakehal Belarbi
Keyword(s):  
Symmetric Spaces ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Riemannian Metric ◽  
Killing Vector Fields ◽  
Generalized Symmetric Space ◽  
Generalized Symmetric Spaces ◽  
Full Classification

In this work we consider the three-dimensional generalized symmetric space, equipped with the left-invariant pseudo-Riemannian metric. We determine Killing vector fields and affine vectors fields. Also we obtain a full classification of Ricci, curvature and matter collineations

scholarly journals ARBTools: A Tricubic Spline Interpolator for Three-Dimensional Scalar or Vector Fields

2019 ◽  
Author(s):  
Paul Walker ◽  
Ulrich Krohn ◽  
Carty David
Keyword(s):  
Magnetic Field ◽  
Vector Field ◽  
Vector Fields ◽  
Three Dimensional ◽  
Spline Method ◽  
Particle Trajectories ◽  
The Core ◽  
Numerical Integrators ◽  
Data Points ◽  
System Requirements

ARBTools is a Python library containing a Lekien-Marsden type tricubic spline method for interpolating three-dimensional scalar or vector fields presented as a set of discrete data points on a regular cuboid grid. ARBTools was developed for simulations of magnetic molecular traps, in which the magnitude, gradient and vector components of a magnetic field are required. Numerical integrators for solving particle trajectories are included, but the core interpolator can be used for any scalar or vector field. The only additional system requirements are NumPy.

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