Homogeneous Vector Fields Research Articles

Some properties of homogeneous vector fields

For planar analytic homogeneous vector fields, the existence of periodic orbits and the nonexistence of limit sets are verified. It is concluded that spacial analytic homogeneous vector field of order m has no limit sets for any m>1. Similar results are extended to higher-dimensional polynomial homogeneous vector fields under certain conditions.

An averaging theorem for time-periodic degree zero homogeneous differential equations

This paper considers the stability of the differential equation dot x = εX(t, x, ε), x ϵ R n , where X( t, x, ε) is a time-periodic, degree zero homogeneous vector field and ε > 0 is a parameter. It is shown that asymptotic stability of the time-averaged equation implies asymptotic stability of the original system for ε sufficiently small.

Projections of Polynomial Vector Fields and the Poincaré Sphere

We investigate projections of homogeneous polynomial vector fields to level sets of homogeneous polynomials. This allows a systematic study of “stationary points at infinity” for polynomial differential equations inn-dimensional real space. Results include some general criteria for the existence of unbounded solutions, and a fairly complete discussion of boundedness of solutions for second-order equations in one dependent variable

Decomposition of homogeneous vector fields of degree one and representation of the flow

We first give a characterization for the set of real analytic diffeomorphisms which transform homogeneous vector fields of certain degree into homogeneous fields of the same degree with respect to an arbitrary dilation δεr. Such a set is constituted by the invertible analytic maps that are homogeneous of degree one with respect to δεr and can be endowed with the structure of a Lie Group whose Lie algebra is the space H1,r(ℝn) of the homogeneous fields of degree one with respect to δεr. Then we prove a decomposition theorem for the elements of the non semisimple Lie algebra H1,r(ℝn). This result is a non linear analog of the Jordan decomposition of a linear field, i.e. for X ∈ H1,r(ℝn), we can write X = S + N, with S linear semisimple and [S, N] = 0. We also give an explicit representation formula for the flow generated by a field in H1,r(ℝn). Finally we apply this result to obtain a simple representation for the trajectories of a class of affine control systems x˙=X0(x)+Bu, with X0 ∈ H1,r(ℝn) and В a constant field, that constitute a natural extension of the linear control systems.

Structural Stability of Planar Homogeneous Polynomial Vector Fields: Applications to Critical Points and to Infinity

LetHmbe the space of planar homogeneous polynomial vector fields of degreemendowed with the coefficient topology. We characterize the setΩmof the vector fields ofHmthat are structurally stable with respect to perturbations inHmand we determine the exact number of the topological equivalence classes inΩm. The study of structurally stable homogeneous polynomial vector fields is very useful for understanding some interesting features of inhomogeneous vector fields. Thus, by using this characterization we can do first an extension of the Hartman–Grobman Theorem which allows us to study the critical points of planar analytical vector fields whosek-jets are zero for allk<munder generic assumptions and second the study of the flows of the planar polynomial vector fields in a neighborhood of the infinity also under generic assumptions.

Algebraic classification of homogeneous polynomial vector fields in the plane

A classification is provided of two-dimensional homogeneous polynomial vector fields of arbitrary degree. This classification results from a consideration of the linear equivalence of such vector fields, using techniques from tensor and spinor algebra. It simplifies, synthesizes and generalizes previous works, which have been restricted to those cases when the polynomials are of no higher than third degree.

The period function of some polynomial systems of arbitrary degree

We consider certain vector fields in the plane which possess a centre. The main result is that for Hamiltonian polynomial systems which are of even degree, which possess homogeneous nonlinearities, and which have a centre located at the origin, the period function is a strictly increasing function of the energy, throughout its interval of definition. It is also shown that for nonlinear homogeneous Hamiltonian polynomial vector fields of arbitrary degree which possess a centre, the period function is a strictly decreasing function of the energy. With appropriate modifications, this result is extended to arbitrary homogeneous vector fields which possess a centre, irrespective of their being Hamiltonian or polynomial; the period function is then strictly monotonic, except when the degree of homogeneity is one, when the systems are isochronous.

Algebraic Conditions for a Centre or a Focus in Some Simple Systems of Arbitrary Degree

Necessary and sufficient conditions are investigated, in the form of explicit algebraic formulas, for the origin to be a centre, an unstable focus, or a stable focus of a homogeneous polynomial planar vector field whose degree is arbitrary. This provides a unified treatment of the problem. The role of these conditions is exemplified in detail in the linear, cubic, and quintic cases. Similar considerations are provided for systems which possess homogeneous non-linearities and for which the origin is a uniformly isochronous centre.

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class L of finite-dimensional, linear, single-input ( u), single-output ( y) systems ( A,b,c) is universally stabilized by the feedback strategy u = Λ( λ) y, λ = y 2, where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class H k of nonlinear control systems ( a,b,c), with positively homogeneous (of degree k ⩾ 1) drift vector field a, is described. Specifically, under the relative-degree-one ( cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ( λ)/ vby/ vb k−1 y, dot λ = /vby/vb k+1 assures H k-universal stabilization. More generally, the strategy u = Λ( λ)exp(/ vby/ vb) y, dot λ = exp(/vby/vb)y 2 assures H -universal stabilization, where H = ∪ k ⩾ 1 H k.

Vectorial calibration of 3D magnetic field sensor arrays

The vectorial calibration of magnetic field sensor arrays along three axes is presented. In this context, a distinction must be made between the calibration of alignment errors between the 3D sensors and the calibration of orthogonality errors between the three axes of a 3D sensor. Alignment errors within the sensor array are determined by using a homogeneous vector field generated by a 3D Helmholtz coil arrangement consisting of single octagonal coils. Two sophisticated methods to determine errors of orthogonality between the axes of a 3D sensor are described and compared. The first attempt was to calibrate the perpendicularity of the 3D coil system by means of a laser-interferometer controlled 90/spl deg/ turn of a solenoid coil and measuring the induced voltage. The second attempt was to use the mechanically fixed perpendicularity errors of the 3D sensor and the 3D coil system. By rotating the 3D sensor in the coil system, the perpendicularity errors of coil system and sensor are superimposed with changing signs and could therefore be calculated. The measurements are taken across a solenoid coil which can be calibrated by the PTB (Physikalisch-Technische Bundesanstalt-Federal Standards Laboratory).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The stabilization of homogeneous cubic vector fields in the plane

In this paper, we deal with the stabilizability of homogeneous polynomials of degree three by means of homogeneous feedbacks of degree two.

Two-Dimensional Homogeneous Polynomial Vector Fields with Common Factors

Invariant characterizations are obtained for the existence of one or more common factors in two-dimensional homogeneous polynomial vector fields of arbitrary degree, r. The presence of a common factor is related to the existence of nonisolated critical points of the vector fields. The particular case where the highest common factor is of maximal degree, r, is studied further, from the invariant point of view. The linear ( r = 1) and quadratic ( r = 2) cases are then examined in the context of the general theory, and the results are contrasted with those which are obtained using more conventional approaches. The relevance of the investigation to certain inhomogeneous polynomial vector fields is briefly discussed. This brings to light an invariant characterization of the classical Jacobi differential equation.

The inverse problem in the Hamiltonian formalism: integrability of linear Hamiltonian fields

The authors study the inverse problem for Hamiltonian dynamics, with restriction to the case of linear homogeneous vector fields. An algebraic necessary and sufficient condition is derived. They study also the problem of symmetries and integrability for a Hamiltonian linear vector field and show that such a field is always completely integrable.

Numerical Simulation Of Homogeneous Non-Gaussian Random Vector Fields

A method which constructs numerical simulations of homogeneous non-Gaussian random vector fields is given. It uses multi-dimensional pulse trains with Poisson arrival times. It is first shown that the classical results concerning one-dimensional filtered Poisson processes can be generalized to the multi-dimensional case. It is then explained how to determine the multi-dimensional filtered Poisson process in order to match the spectral density matrix and the first statistical moments of the given non-Gaussian random field. This method is used in order to construct non-Gaussian simulation of wind. Finally, the sensitivity of the response of a non-linear dynamical system excited by a random process to the input law is illustrated by an example.

Homogeneous Lyapunov function for homogeneous continuous vector field

The goal of this article is to provide a construction of a homogeneous Lyapunov function V associated with a system of differential equations x dot f(x), x ϵ R n (n≥1) , under the hypotheses: (1) f ϵ C( R n, R n) vanishes at x = 0 and is homogeneous; (2) the zero solution of this system is locally asymptotically stable. Moreover, the Lyapunov function V(x) tends to infinity with ‖x‖, and belongs to C ∞( R n/{0}, R)∩C p( R n, R) , with pϵ N ∗ as large as wanted. As application to the theory of homogeneous systems, we present two well known results of robustness, in a slightly extended form, and with simpler proofs.

Genericity of hyperbolic homogeneous vector fields in R 3

Genericity of hyperbolic homogeneous vector fields in R 3

Stochastic FEM Based on Local Averages of Random Vector Fields

The formula for the covariances of the local averages of homogeneous random scalar fields over rectangular domains is first generalized to include the case of homogeneous random vector fields with quadrant symmetry. For nonhomogeneous random fields and/or nonrectangular domains, Gaussian quadrature is proposed to evaluate the means and covariances of the local averages of random vector fields. The stochastic finite‐element analysis based on the local averages of random vector fields is then formulated for static, eigenvalue, and stress‐intensity factor problems. The present stochastic finite‐element method (SFEM) is a generalization of the SFEM based on the local averages of random scalar fields and can be applied to any configuration of structure with several correlated random parameters and several correlated random loads. Numerical examples are examined to show the advantages of the present SFEM over the SFEM based on midpoint discretization, i.e., more rapid convergence and more efficiency.

Transition functions and moduli of stability for 3-dimensional homogeneous vector fields with a hyperbolic blowing-up

In the first part of the paper we introduce the “Normal transition function” for saddle connections of planar diffeomorphisms. It is a positive multiple of the usual transition function, but in its definition we do not need C 1 -linearizing coordinates. Among other nice properties, it is found to be analytic when the diffeomorphism is. The second part of the paper deals with the existence of a modulus of stability for germs on R 3 of homogeneous vector fields with a hyperbolic blowing-up. We show that inside a specific class of examples the modulus occurs for a sufficiently high degree.

Kinematic modeling of cross‐sectional deformation sequences by computer simulation

We derive a kinematic algorithm which permits simulation of postulated cross‐sectional deformation sequences in sedimentary rocks affected by faulting, fault‐related folding, and simple shear. The transformations are to a more deformed state and are formulated analytically in terms of the less deformed configuration (forward modeling, Lagrangian description). The medium is subdivided into domains of constant dip and homogeneous displacement vector fields which are delimited by the axial planes of the fault inflections. The displacement trajectory is of constant length for all the displaced particles throughout the medium and parallel to the underlying active fault segment. As a consequence, the deformation path is continuous but not smooth and causes an angular style of parallel folding, except at the deformation front where the layer thickness is not conserved. The inhomogeneity of the displacement vector field across axial planes introduces longitudinal and angular shear strains, whereas the area of the medium remains constant. These strains, which are superposed on the externally applied simple shear, make the mapping transformation unconformal. We define the stratigraphic layering of the undeformed medium by an approximating body of finite quadrilateral elements, evaluate the defined displacement functions for the grid nodes, and graph the deformed mesh. Eventually, we create with the algorithm synthetic deformation sequences for simple boundary conditions and try in a specific case study to match the resulting synthetics with the available observational and experimental data.

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

Tsutomu Date and Masao Iri in [DI] gave an algebraic classification of systems I = P(x, y), I; = Q(.x, y), where P and Q are homogeneous polynomials of degree 2. For this, they used the classification of the binary cubic forms and also the simultaneous classification of a linear binary form and a cubic binary form given by the algebraic invariant theory. We begin by doing a similar study for systems .?? = P(x, y), ,’ = Q(x, y), where P and Q are homogeneous polynomials of degree three (i.e., cubic systems). The classification’s theorem of such systems is based on the classification of fourth-order binary forms. Gurevich in [Gu] did the classification of fourth-order binary forms on the field of complex numbers. Since we did not find the classification on the real domain, we adapt Gurevich’s proof to obtain it. In Section 1 we give some definitions and preliminary results, while in Section 2 we give the theorem of classification of fourth-order binary forms on the real domain. The method used in the proof (Caley’s method) let us obtain canonical forms of the fourth-order binary forms and the algebraic characteristics. Section 3 is devoted to obtaining the algebraic classification of systems .? = P(x, y), j = Q(x, y), where P and Q are homogeneous polynomials of degree 3. Given an arbitrary system X= (P, Q) with P and Q homogeneous polynomials of degree 3, we can know the equivalence-class at which it belongs through the algebraic characteristics. In Section 4 we study the phase-portraits of systems 1= P(x, y), j = Q(x, y), where P and Q are homogeneous polynomials of degree n and P and Q have no common factor. Such systems have been studied by J. Argemi in [A]. Here we give a shorter new proof of his results by using