Homogeneous Vector Fields Research Articles

A Neural Network Approach to Finding Global Lyapunov Functions for Homogeneous Vector Fields

A Neural Network Approach to Finding Global Lyapunov Functions for Homogeneous Vector Fields

Homogeneous systems stabilization based on convex embedding

The paper develops control algorithms for a class of affine nonlinear systems using the so-called canonical homogeneous representation. It is demonstrated that such a representation exists for any homogeneous vector field bounded on the unit sphere. It is shown that canonical homogeneous representation is useful for LMI-based control design and stability analysis of nonlinear systems. Theoretical results are supported by numerical examples.

A novel eigenvalue-based iterative simulation method for multi-dimensional homogeneous non-Gaussian stochastic vector fields

The generation of sample functions of stochastic fields or processes is a prerequisite to use the Monte Carlo method in stochastic mechanics and structural reliability analysis. When it comes to non-Gaussian characteristics such as the spatial variability of material or structural properties, the simulation of multi-dimensional non-Gaussian stochastic vector fields becomes necessary. However, the conventional Cholesky decomposition-based iterative simulation method requires three times of decomposition operations at each iteration and a complex procedure of random shuffle, which inevitably augment the model complexity. To this end, this study develops a novel eigenvalue-based iterative simulation method which only requires one decomposition operation in the whole simulation process. Central to this method is to represent the cross power spectral density matrix with its eigenvalue matrix during the iteration procedure, which avoids the cumbersome Cholesky decomposition and random shuffle procedure. Meanwhile, it can yield unique convergence result and present a powerful ability in addressing cases with the complex-valued spectral matrix. Numerical example demonstrates that the proposed method has a similar accuracy as that of the Cholesky decomposition-based iterative simulation method. Meanwhile, the estimated cumulative distribution function from one simulated sample and the estimated ensemble correlation function from 1000 simulated samples are all consistent with their respective targets. In addition, the sensitive analysis is performed to demonstrate that the proposed method has a relatively wide applicability. Therefore, this method may have a great potential to simulate multi-dimensional homogeneous non-Gaussian stochastic vector fields in practice.

On finite‐time stability analysis of homogeneous vector fields with multiplicative perturbations

AbstractThe problem of finite‐time and fixed‐time stability analysis is considered for a class of nonlinear systems where is homogeneous of a degree and is bounded. It is shown that under certain conditions on the asymptotic stability of the system implies its finite‐time stability for or nearly fixed‐time stability for . The so‐called homogeneous extensions are utilized for the analysis. The results are generalized to a class of systems where for every are homogeneous of some (possibly different) degrees and satisfy certain boundedness conditions. An example of a mechanical system is presented in the last section to illustrate the obtained results.

Mass under the Membrane Theory of Gravitation

The Cosmic Membrane theory of gravitation (CM) implies Newton’s absolute space. We identify the homogeneous vector field used by us since 1994 with the Higgs-field as source of the heavy mass. Following Randall and Sundrum, the introduction of the wafting layer outside the membrane solves the issue of the mobility of particles in a super-strong membrane. Starting with Feynman’s radius of excess, we obtain a depth of space of WRS = 1.432×106 [m] of the gravitational funnel at the edge of sun. Using Chandrasekhar’s gravitational energy, we obtain the tension F0 of the membrane as F0=1.820×1019 [N/m2], and the vertical vector field acceleration AVFV, acting perpendicularly from the fourth spatial dimension on the membrane, with AVFV=1.148×105 [m/s2]. The horizontal vector field acceleration AVFH, i.e., inside the wafting layer, is AVFH=1.330×105 [m/s2], and acts as acceleration a=AVFH w’ with w’ the being slope of the membrane. The mass of the moved membrane in a moving gravitational funnel behaves as an inert mass, but yields a numerical value that is too small to explain the equivalence of heavy and inert mass. Assuming speed of light c for transversal gravitational waves, we obtain a first estimation of the mass distribution ρsurf of the membrane. The clay lump model of the relativistic increase of mass follows the assumption that the energy of the accelerating photons will act again half as mass and half as kinetic energy at the accelerated particle. Our result equals exactly Einstein’s SR result.

Hamiltonian nilpotent saddles of linear plus cubic homogeneous polynomial vector fields

We completely characterize the global phase portraits in the Poincaré disk for all planar Hamiltonian vector fields with linear plus cubic homogeneous terms having a nilpotent saddle at the origin.

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.

Depth of Space and Radius of the Electron

The cosmological model of the expanding balloon in 4D-space (CM) delivers in interaction with a homogeneous vector field exactly Newton’s law of gravitation with its 1/r-shape of the gravitational funnel. So far, the depth of space, W, in the 4-th spatial dimension can only be calculated using the theoretical approach of Feynman’s radius of excess rex=a/3 with Schwarzschild-radius a. With this, the connection to the general theory of relativity (GR) is established, but the situation is unsatisfactory. In the present study, the possibilities of an experimental approach to the calculation of spatial depth, W, are explored. The only experimental approach so far is the bending of light on a central mass. We hypothesize in addition to the main effect φ = -4a/y, i.e., the angle of diffraction of a light beam on a heavy central mass in the distance y and with Schwarzschild-radius a, an additional effect close to the center of the form φC ~ -1/y4. This additional effect has on the edge of the central mass about 1/3 of the strength of the main effect. However, its influence disappears very quickly with increasing distance. For this reason the sun cannot be used as the central mass. The bright corona and the strong magnetosphere do not allow measurements close to the sun. However, ESA’s GAIA mission puts the planet Jupiter at the center of interest. This spacecraft measures with extremely high precision the positions of billions of stars. Results of first data analyses have already been published. As a side effect - the application of the CM to small particles provides an indication that the radius of the electron could be in the order of 10-23 m.

Algebraic and topological classification of homogeneous quartic vector fields in the plane

We provide canonical forms for the homogeneous polynomials of degree five. Then we characterize all the phase portraits in the Poincare disk for all quartic homogeneous polynomial differential systems. More precisely, there are exactly 23 different topological phase portraits for the quartic homogeneous polynomial differential systems.

Limit cycles for some families of smooth and non-smooth planar systems

We apply the averaging method in a class of planar systems given by a linear center perturbed by a sum of continuous homogeneous vector fields, to study lower bounds for their number of limit cycles. Our results can be applied to models where the smoothness is lost on the set Σ={xy=0}. They also motivate to consider a variant of Hilbert 16th problem, where the goal is to bound the number of limit cycles in terms of the number of monomials of a family of polynomial vector fields, instead of doing this in terms of their degrees.

Formula omitted]-stability of generalized homogeneous cooperative systems

We first propose a class of generalized homogeneous vector fields which include existing ones as special cases. Then we establish a necessary and sufficient condition for D-stability of more general homogeneous cooperative systems. The decay rate is also explicitly estimated. Two numerical examples are worked out to show the effectiveness of the main result.

Global estimates in Sobolev spaces for homogeneous Hörmander sums of squares

Let L=∑j=1mXj2 be a Hörmander sum of squares of vector fields in space Rn, where any Xj is homogeneous of degree 1 with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for L in the X-Sobolev spaces WXk,p(Rn), where X={X1,…,Xm}. In our approach, we combine local results for general Hörmander sums of squares, the homogeneity property of the Xj's, plus a global lifting technique for homogeneous vector fields.

Phase Portraits of Random Planar Homogeneous Vector Fields

We study the phase portraits with positive probability of random planar homogeneous vector fields of degree n. In particular, for n=1,2,3, we give a complete solution of the problem and, moreover, either we give the exact value of each probability or we estimate it by using the Monte Carlo method. It is remarkable that all but two of these phase portraits are characterized by their index at the origin and by their number of invariant straight lines through it.

Pohozaev-type identities for differential operators driven by homogeneous vector fields

We prove Pohozaev-type identities for smooth solutions of Euler-Lagrange equations of second and fourth order that arise from functional a depending on homogeneous Hörmander vector fields. We then exploit such integral identities to prove non-existence results for the associated boundary value problems.

Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions

We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Lévy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Lévy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence.

Flow of Spatiotemporal Turbulentlike Random Fields

We study the Lagrangian trajectories of statistically isotropic, homogeneous, and stationary divergence-free spatiotemporal random vector fields. We design this advecting Eulerian velocity field such that it gets asymptotically rough and multifractal, both in space and time, as it is demanded by the phenomenology of turbulence at infinite Reynolds numbers. We then solve numerically the flow equations for a differentiable version of this field. We observe that trajectories get also rough, characterized by nearly the same Hurst exponent as the one of our prescribed advecting field. Moreover, even when considering the simplest situation of the advection by a fractional Gaussian field, we evidence in the Lagrangian framework additional intermittent corrections. The present approach involves properly defined random fields, and asks for a rigorous treatment that would explain our numerical findings and deepen our understanding of this long lasting problem.

Poincaré Compactification for Non-polynomial Vector Fields

In this work a theorical framework to apply the Poincare compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincare compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.

Regularisation for Planar vector fields

This paper serves as a first foray on regularisation for planar vector fields. Motivated by singularities in celestial mechanics, the block regularisation of a generic class of degenerate singularities is studied. The paper is concerned with asymptotic properties of the transition map between a section before and after the singularity. Block regularisation is reviewed before topological and explicit conditions for the C0-regularity of the map are given. Computation of the C1-regularisation is reduced to summing residues of a rational function. It is shown that the transition map is in general only finitely differentiable and a method of computing the map is conveyed. In particular, a perturbation of an example derived from the four-body problem is shown to be C4/3. The regularisation of all homogeneous quadratic vector fields is computed.

On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of $\mathbb{P}^2$ and quadratic homogeneous vector fields on $\mathbb{C}^3$, the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on $\mathbb{C}^3$ having exclusively single-valued solutions.

An application of a global lifting method for homogeneous Hörmander vector fields to the Gibbons conjecture

In this paper we exploit a global lifting method for homogeneous Hormander vector fields in order to extend the Gibbons conjecture to any second-order differential operator $$\mathcal {L}_X = \sum _{j = 1}^mX_j^2$$, where the $$X_j$$’s are linearly independent smooth vector fields on $$\mathbb {R}^n$$ satisfying Hormander’s rank condition and fulfilling a suitable homogeneity property with respect to a family of non-isotropic dilations. The class of these operators comprehends the sub-Laplacians on Carnot groups, the smooth Grushin-type operators and the smooth $$\Delta _\lambda $$-Laplacians studied by Franchi, Lanconelli and Kogoj. We also establish a comparison result for the solutions of the semi-linear equation $$\mathcal {L}_Xu+f(u) = 0$$ under suitable assumptions on the function f.