Topological Equivalence of Dicritical Holomorphic Vector Fields

In this paper we study the topological equivalence between holomorphic vector fields with dicritical isolated singularity and without singularities at the divisor.     

Sphere Bundles Transverse to Holomorphic Vector Fields

We prove that for every pair of nonzero complex numbers λ ₁ and λ ₂ with \(\frac {\lambda _{1}}{\lambda _{2}}\not \in \mathbb {R}\) there is an embedding \(S^{2}\times S^{1}\rightarrow \mathbb {C}^{2}\) transverse to the linear holomorphic vector field \(Z(x,y)=\lambda _{1}x\frac {\partial }{\partial x}+\lambda _{2} y\frac {\partial }{\partial y}\) . This extends a previous result by Ito (1989).     

Rigidity Theorem for Degenerated Singular Points of Germs of Holomorphic Vector Fields in the Complex Plane

We consider the class _n of germs of holomorphic vector fields in ( , 0) with vanishing (n – 1)-jet at zero. We prove that the formal equivalence of two generic germs in _n implies their analytic equivalence. This result is analogous to the one obtained in [16] for the case of orbital analytic equivalence.     

Twin Vector Fields and Independence of Spectra for Quadratic Vector Fields

The object of this paper is to address the following question: when is a polynomial vector field on \(\mathbb {C}^{2}\) completely determined (up to affine equivalence) by the spectra of its singularities? We will see that for quadratic vector fields, this is not the case: given a generic quadratic vector field there is, up to affine equivalence, exactly one other vector field which has the same spectra of singularities. Let us say that two distinct vector fields are twin vector fields if they have the same singular locus and the same spectrum at each singularity. Our main result is as follows: any two generic quadratic vector fields with the same spectra of singularities (yet possibly different singular locus) can be transformed by suitable affine maps to be either the same vector field or a pair of twin vector fields. Moreover, a generic quadratic vector field has exactly one twin vector field. We later analyze the case of quadratic Hamiltonian vector fields in more detail and find necessary and sufficient conditions for a collection of non-zero complex numbers to arise as the spectra of singularities of a quadratic Hamiltonian vector field. Lastly, we show that a generic quadratic vector field is completely determined (up to affine equivalence) by the spectra of its singularities together with the characteristic numbers of its singular points at infinity.     

Nonoscillating Projections for Trajectories of Vector Fields

The properties of oscillation for trajectories of three-dimensional vector fields are “visible” in the sense that they can be detected by means of plane projections. We also recover a three-dimensional version of the classical dichotomy “nonoscillation” versus “spiralling” for plane vector fields.      

The Extended Monodromy Group and Liouvillian First Integrals

We present a connection between the differential Galois theory and topological Galois theory. The first one is represented by Singer's theorem about integration of polynomial planar vector fields and the second is Khovanskii's monodromy group of a multivalued analytic first integral of such vector field. We show that the monodromy maps together with certain additional maps, which are analogues of the Stokes operators, generate a solvable group.     

Limit Cycles for Multidimensional Vector Fields. The Elliptic Case

We consider polynomial vector fields of the form and their polynomial perturbations of degree n. We present a sufficient condition that the perturbed system has an invariant surface close to the plane z = 0. We study limit cycles which appear on this surface. The linearized condition for limit cycles, bifurcating from the curves y ² – x ³ + 3x = h, leads to a certain 2- dimensional integral (which generalizes the elliptic integrals). We show that this integral has a representation R ₁(h)I ₁ + + R _e(h)I _e, where R _j (h) are rational functions with degrees of numerators and denominators bounded by O(n). In the case of constant and one-dimensional matrix A(x,y) we estimate the number of zeros of the integral by const n.     

Analytical Classification of Singular Saddle-Node Vector Fields

We consider the set Z_SNof germs of holomorphic saddlenode (or degenerate resonant) vector fields at the origin of C²and study the local analytical classi.cation, i.e., describing the set of orbits of Diiff(C², 0) acting by change of coordinates on Z_SN. J. Martinet and J.-P. Ramis studied this classi.cation problem in the case of the underlying foliation, leading to functional invariants. When two vector fields induce the same foliation, we show that conjugating them is equivalent to solving some homological equation associated with one of these vector fields. Therefore, we can read the existence of such a change of coordinates in some integrals along asymptotic cycles of the foliation, providing a complete set of functional invariants in addition to Martinet—Ramis invariants.     

On Curvatures and Focal Points of Distributions of Dynamical Lagrangian Distributions and their Reductions by First Integrals

Pairs (Hamiltonian system, Lagrangian distribution) called dynamical Lagrangian distributions, appear naturally in differential geometry, calculus of variations, and rational mechanics. The basic differential invariants of a dynamical Lagrangian distribution with respect to the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and curvature form. These invariants can be considered as generalizations of the classical curvature tensor in Riemannian geometry. In particular, in terms of these invariants one can localize the focal points along extremals of the corresponding variational problems. In the present paper we study the behavior of the curvature operator, the curvature form, and the focal points of a dynamical Lagrangian distribution after its reduction by arbitrary first integrals in involution. An interesting phenomenon is that the curvature form of so-called monotone increasing Lagrangian dynamical distributions, which appear naturally in mechanical systems, does not decrease after reduction. It also turns out that the set of focal points to the given point with respect to the monotone increasing dynamical Lagrangian distribution and the corresponding set of focal points with respect to its reduction by one integral are alternating sets on the corresponding integral curve of the Hamiltonian system of the considered dynamical distributions. Moreover, the first focal point corresponding to the reduced Lagrangian distribution comes before any focal point related to the original dynamical distribution. We illustrate our results on the classical N-body problem. 2000 Mathematics Subject Classification. 37J15, 37J05, 53D20.     

Normalizability, Synchronicity, and Relative Exactness for Vector Fields in C2

In this paper, we study the necessary and su.cient condition under which an orbitally normalizable vector field of saddle or saddle-node type in C²is analytically conjugate to its formal normal form (i.e., normalizable) by a transformation fixing the leaves of the foliation locally.First, we express this condition in terms of the relative exactness of a certain 1-form derived from comparing the time-form of the vector field with the time-form of the normal form. Then we show that this condition is equivalent to a synchronicity condition: the vanishing of the integral of this 1-form along certain asymptotic cycles de.ned by the vector field. This can be seen as a generalization of the classical Poincaré theorem saying that a center is isochronous (i.e., synchronous to the linear center) if and only if it is linearizable.The results, in fact, allow us in many cases to compare any two vector fields which differ by a multiplicative factor. In these cases we show that the two vector fields are analytically conjugate by a transformation fixing the leaves of the foliation locally if and only if their time-forms are synchronous.     

On the Stokes Phenomenon for Holomorphic Solutions of Integro-Differential Equations with Irregular Singularity

We study the Stokes phenomenon for sectorial holomorphic solutions of linear integro-differential equations in two variables t, z with irregular singularity at t = 0. More precisely, we give sufficient conditions on the coefficients of the equations and initial conditions under which the Stokes transition functions can be expressed as the generalized Laplace transform of a convergent series of hyperfunctions defined on half-lines in .      

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n?1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem.     

Accurate Computation of Axisymmetric Vector Potential Fields with the Finite Element Method

A highly efficient formulation is described for implementing the finite element method for axisymmetric vector field analysis. The method was tested to investigate its full potential for the solution of steady-state and transient electromagnetic fields before application to specific problems. Computational tests show that the method requires comparable computing time to the equivalent Cartesian coordinate problem. The results of some numerical examples are presented including problems with different time-dependent source vectors. Results obtained are compared, where possible, against analytical solutions, as well as experimental results. An error analysis confirms that the accuracy is very good even for elements placed near the axis. Hence the singularity problem is effectively removed, as expected.     

Nonlinear Vector Potential Formulation and Experimental Verification of Newton-Raphson Solution of Three Dimensional Magnetostatic Fields in Electrical Devices

An iterative technique, based on magnetic vector potential formulation and the Newton-Raphson method, for the determination of the three dimensional magnetostatic field distributions in electrical devices is given. The proper degrees of magnetic saturation in the various materials within a given volume under consideration are obtained by repeated evaluation of the reluctivities in that volume, using a cubic spline representation of the B-H magnetization characteristics of composite materials (laminations). The formulation has been applied to a practical example of determining the field in and around a shell type 1.5 kva single phase transformer. The convergence and implementation characteristics of the developed method are given in this paper which show a saving of about 34% in CPU solution time in comparison with previously published methods. Experimental verification is given in terms of a comparison between computed and experimentally obtained values of flux densities surrounding the transformer core and winding, under heavily saturated conditions. Excellent agreement between test and calculated flux densities was achieved. This method is thus quite applicable to the solution of a wide class of three dimensional magnetostatic field problems associated with electrical apparatus.     

Application of Physical Fields for Increasing Oil Recovery Efficiency

The effective factor of regulating of rheophysical characteristics of liquid systems in oil recovery is use of physical and physical-chemical methods. The convincing outcomes of applying of physical-magnetic and electric fields are obtained on thermodynamic properties of liquid systems in processes of oil recovery, cleaning of propellant and oils, etc. Results of theoretical, experimental, and field studies and applying of the new methods of increasing of oil recovery efficiency by treating of aqueous systems by electric and magnetic fields are observed.     

黑麋峰抽水蓄能电站首台机组首次起动方式分析

结合黑麋峰抽水蓄能电站的实际特点,深入分析各种不同起动方式下可能产生的影响因素、合理选择首台机组首次起动方式,以确保合理可行的水库蓄水计划安排及机组安全、稳定、经济运行。     

黑麇峰抽水蓄能电站首台机组首次起动方式分析

结合黑麋峰抽水蓄能电站的实际特点,深入分析各种不同起动方式下可能产生的影响因素、合理选择首台机组首次起动方式,以确保合理可行的水库蓄水计划安排及机组安全、稳定、经济运行.