Narrow Results

If $L$ is a semisimple Lie algebra of vector fields on ${ℝ}^N$ with a split Cartan subalgebra $C$, then it is proved here that the dimension of the generic orbit of $C$ coincides with the dimension of $C$. As a consequence one obtains a local canonical form of $L$ in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates — for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on ${ℝ}^3$ and to determine all representations of $\mathrm{\text{sl}}(N,ℝ)$ as local vector fields on ${ℝ}^{N-1}$. These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits $\mathrm{\text{sl}}(3,ℝ)$ as its symmetry algebra and for a system of two second-order ordinary differential equations that admits $\mathrm{\text{sl}}(4,ℝ)$ as its symmetry algebra.

We consider Faddeev formulation of General Relativity (GR) in which the metric is composed of ten vector fields or a 4 ×10 tetrad. This formulation reduces to the usual GR upon partial use of the field equations.

A distinctive feature of the Faddeev action is its finiteness on the discontinuous fields. This allows to introduce its minisuperspace formulation where the vector fields are constant everywhere on ℝ⁴ with exception of a measure zero set (the piecewise constant fields). The fields are parametrized by their constant values independently chosen in, e.g. the 4-simplices or, say, parallelepipeds into which ℝ⁴ can be decomposed. The form of the action for the vector fields of this type is found.

We also consider the piecewise constant vector fields approximating the fixed smooth ones. We check that if the regions in which the vector fields are constant are made arbitrarily small, the minisuperspace action and equations of motion tend to the continuum Faddeev ones.

The inflaton coupling to a vector field via the $f{(\phi )}^2{F}_{\mu \nu }{F}^{\mu \nu }$ term is used in several contexts in the literature, such as to generate primordial magnetic fields, to produce statistically anisotropic curvature perturbation, to support anisotropic inflation, and to circumvent the $\eta$-problem. In this work, I perform dynamical analysis of this system allowing for the most general Bianchi I initial conditions. I also confirm the stability of attractor fixed points along phase–space directions that had not been investigated before.

We construct examples of vector fields on a three-sphere, amenable to analytic proof of properties that guarantee the existence of complex behavior. The examples are restrictions of symmetric polynomial vector fields in R⁴ and possess heteroclinic networks producing switching and nearby suspended horseshoes. The heteroclinic networks in our examples are persistent under symmetry preserving perturbations. We prove that some of the connections in the networks are the transverse intersection of invariant manifolds. The remaining connections are symmetry-induced. The networks lie in an invariant three-sphere and may involve connections exclusively between equilibria or between equilibria and periodic trajectories. The same construction technique may be applied to obtain other examples with similar features.

In this paper we study C¹ linearization and classification of germs of hyperbolic vector fields on Rⁿ. Two sets of algebraic conditions imposed on the eigenvalues of vector fields are given, under either of which strict similarity of the linear parts is necessary and sufficient for C¹ conjugacy of two vector fields. Furthermore, one set of the algebraic conditions together with strict similarity also imply linearization.

We study autonomous systems of first order ordinary differential equations, their corresponding vector fields and the autonomous system corresponding to the vector field of the commutator of two such autonomous systems. These vector fields form a Lie algebra. From the variational equations of these autonomous systems, we form new vector fields consisting of the sum of the two vector fields. We show that these new vector fields also form a Lie algebra. Results about fixed points, first integrals and the divergence of the vector fields are also presented.

Noncommutative derivative operators acting on the quantum 3D space in the sense of Manin are introduced. Furthermore, the quantum 3D space is extended by the series expansion of the logarithm of the grouplike generator in the quantum 3D space. We give its differential calculus and the corresponding Weyl algebra. We also obtain algebra of Cartan–Maurer forms on this extension and the corresponding Lie algebra of vector fields. All noncommutative results are found to reduce to those of the standard commutative algebra when the deformation parameter of the quantum 3D space is set to one.

This paper considers the possibility that abstract thinking and advanced synthesis skills might encourage extraterrestrial civilizations to accept communication with mankind on Earth. For this purpose, a notation not relying upon the use of alphabet and numbers is proposed, in order to denote just some basic geometric structures of current physical theories: vector fields, $1$-form fields, and tensor fields of arbitrary order. An advanced civilization might appreciate the way here proposed to achieve a concise description of electromagnetism and general relativity, and hence it might accept the challenge of responding to our signals. The abstract symbols introduced in this paper to describe the basic structures of physical theories are encoded into black and white bitmap images that can be easily converted into short bit sequences and modulated on a carrier wave for radio transmission.

It is well known that the celebrated Lorenz system has an attractor such that every orbit ends inside a certain ellipsoid in forward time. We complement this result by a new phenomenon and by a new interpretation. We show that "infinity" is a global repeller for a set of parameters wider than that usually treated. We construct in a compacted space, a unit sphere that serves as the image of an ideal set at infinity. This sphere is shown to be the union of a family of periodic solutions. Each periodic solution is viewed as a limit cycle, or an isolated periodic orbit when restricted to a certain plane. The unconventional compactification is used.

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations ẏ(t) = g(y(t)) on ℝ^d and those of the parabolic equations on a bounded domain Ω. We give details on the similarities of these dynamics in the cases d = 1, d = 2 and d ≥ 3 and in the corresponding cases Ω = (0, 1), Ω = 𝕋¹ and dim(Ω) ≥ 2 respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations.

In this article some qualitative aspects of non-smooth systems on ℝⁿ are studied through methods of Geometric Singular Perturbation Theory (GSP-Theory). We present some results that generalize some settings in low dimension, that bridge the space between such systems and singularly perturbed smooth systems. We analyze the local behavior around typical singularities and prove that the dynamics of the so called Sliding Vector Field is determined by the reduced problem on the center manifold.

A class of equations describing the geodesic flow for a right-invariant metric on the group of diffeomorphisms of ℝⁿ is reviewed from the viewpoint of their Lie-Poisson structures. A subclass of these equations is analogous to the Euler equations in hydrodynamics (for n = 3), preserving the volume element of the domain of fluid flow. An example in n = 1 dimension is the Camassa-Holm equation, which is a geodesic flow equation on the group of diffeomorphisms, preserving the H¹ metric.

We introduce a notion of convexity in the geometry of vector fields and we prove a PDE-characterization for such notion and related properties.

A simple Lagrangian (with squared covariant divergence of a vector field as a kinetic term) turned out an adequate tool for macroscopic description of dark sector. The zero-mass field acts as the dark energy. Its energy-momentum tensor is a simple additive to the cosmological constant. Space-like and time-like massive vector fields describe two different forms of dark matter. The space-like field is attractive. It is responsible for the observed plateau in galaxy rotation curves. The time-like massive field displays repulsive elasticity. In balance with dark energy and ordinary matter it provides a four-parametric diversity of regular solutions of the Einstein equations describing different possible cosmological and oscillating non-singular scenarios of evolution of the Universe. In particular, the singular “big bang” turns into a regular inflation-like transition from contraction to expansion with accelerated expansion at late times. The fine-tuned Friedman-Robertson-Walker singular solution is a particular limiting case at the boundary of existence of regular oscillating solutions (in the absence of vector fields). The simplicity of the general covariant expression for the energy-momentum tensor allows analyzing the main properties of the dark sector analytically, avoiding unnecessary model assumptions.